Get answer: Using Properties of Definite Integrals Given and evaluate (a) (b) (c) (d)

In Exercises 1–6, find the indefinite integral \(\int\left(4 x^{2}+x+3\right) d x\) Text Transcription: int(4 x^{2}+x+3) d x

In Exercises 1–6, find the indefinite integral. \(\int \frac{6}{\sqrt[3]{x}} d x\) Text Transcription: \int \frac{6} \sqrt 3{x} d x

In Exercises 1–6, find the indefinite integral. \(\int \frac{x^{4}+8}{x^{3}} d x\) Text Transcription: \int \frac x^4+8 x^3 d x

In Exercises 1–6, find the indefinite integral. \(\int\left(5 \cos x-2 \sec ^{2} x\right) d x\) Text Transcription: \int (5 \cos x-2 \sec ^{2} x) d x

In Exercises 1–6, find the indefinite integral. \(\int\left(5-e^{x}\right) d x\) Text Transcription: \int (5-e^x) d x

In Exercises 1–6, find the indefinite integral. \(\int \frac{10}{x} d x\) Text Transcription: \int \frac{10}{x} d x

In Exercises 7–10, find the particular solution that satisfies the differential equation and the initial condition. \(f^{\prime}(x)=-6 x, f(1)=-2\) Text Transcription: f^{\prime}(x)=-6 x, f(1)=-2

In Exercises 7–10, find the particular solution that satisfies the differential equation and the initial condition. \(f^{\prime}(x)=9 x^{2}+1, f(0)=7\) Text Transcription: f^\prime(x)=9 x^{2}+1, f(0)=7

In Exercises 7–10, find the particular solution that satisfies the differential equation and the initial condition. \(f^{\prime \prime}(x)=24 x, f^{\prime}(-1)=7, f(1)=-4\) Text Transcription: f^{\prime \prime}(x)=24 x, f^{\prime}(-1)=7, f(1)=-4

In Exercises 7–10, find the particular solution that satisfies the differential equation and the initial condition. \(f^{\prime \prime}(x)=2 \cos x, f^{\prime}(0)=4, f(0)=-5\) Text Transcription: f^\prime \prime(x)=2 \cos x, f^\prime(0)=4, f(0)=-5

A ball is thrown vertically upward from ground level with an initial velocity of 96 feet per second. Use a(t) =-32 feet per second per second as the acceleration due to gravity. (Neglect air resistance.) (a) How long will it take the ball to rise to its maximum height? What is the maximum height? (b) After how many seconds is the velocity of the ball one-half the initial velocity? (c) What is the height of the ball when its velocity is one-half the initial velocity?

The speed of a car traveling in a straight line is reduced from 45 to 30 miles per hour in a distance of 264 feet. Find the distance in which the car can be brought to rest from 30 miles per hour, assuming the same constant deceleration

In Exercises 13 and 14, find the sum. Use the summation capabilities of a graphing utility to verify your result. \(\sum_{i=1}^{5}(5 i-3)\) Text Transcription: \sum_{i=1}^{5}(5 i-3)

In Exercises 13 and 14, find the sum. Use the summation capabilities of a graphing utility to verify your result. \(\sum_{k=0}^{3}\left(k^{2}+1\right)\) Text Transcription: \sum_{k=0}^{3}\left(k^{2}+1\right)

In Exercises 15 and 16, use sigma notation to write the sum. \(\frac{1}{3(1)}+\frac{1}{3(2)}+\frac{1}{3(3)}+\cdots+\frac{1}{3(10)}\) Text Transcription: \frac 1 3(1) +\frac 1 3(2)+\frac 13(3)+\cdots+\frac 1 3(10)

In Exercises 15 and 16, use sigma notation to write the sum. \(\left(\frac{3}{n}\right)\left(\frac{1+1}{n}\right)^{2}+\left(\frac{3}{n}\right)\left(\frac{2+1}{n}\right)^{2}+\cdots+\left(\frac{3}{n}\right)\left(\frac{n+1}{n}\right)^{2}\) Text Transcription: (\frac{3}{n})(\frac{1+1}{n})^{2}+(\frac{3}{n})(\frac{2+1}{n})^{2}+\cdots+(\frac{3}{n})(\frac n+1 n)^2

In Exercises 17–20, use the properties of summation and Theorem 5.2 to evaluate the sum. \(\sum_{i=1}^{20} 2 i\) Text Transcription: \sum_{i=1}^{20} 2 i

In Exercises 17–20, use the properties of summation and Theorem 5.2 to evaluate the sum. \(\sum_{i=1}^{30}(3 i-4)\) Text Transcription: \sum_{i=1}^{30}(3 i-4)

In Exercises 17–20, use the properties of summation and Theorem 5.2 to evaluate the sum. \(\sum_{i=1}^{20}(i+1)^{2}\) Text Transcription: \sum_{i=1}^{20}(i+1)^{2}

In Exercises 17–20, use the properties of summation and Theorem 5.2 to evaluate the sum. \(\sum_{i=1}^{12} i\left(i^{2}-1\right)\) Text Transcription: \sum_{i=1}^{12} i(i^{2}-1)

In Exercises 21–24, use the limit process to find the area of the region bounded by the graph of the function and the x-axis over the given interval. Sketch the region. y =8-2x, [0, 3]

In Exercises 21–24, use the limit process to find the area of the region bounded by the graph of the function and the -axis over the given interval. Sketch the region. \(y=x^{2}+3, \quad[0,2]\) Text Transcription: y=x^{2}+3, \quad[0,2]

In Exercises 21–24, use the limit process to find the area of the region bounded by the graph of the function and the -axis over the given interval. Sketch the region. \(y=5-x^{2}, \quad[-2,1]\) Text Transcription: y=5-x^{2}, \quad[-2,1]

In Exercises 21–24, use the limit process to find the area of the region bounded by the graph of the function and the -axis over the given interval. Sketch the region. \(y=\frac{1}{4} x^{3}, \quad[2,4]\) Text Transcription: y=\frac{1}{4} x^{3}, \quad[2,4]

Use the limit process to find the area of the region bounded by \(x=5 y-y^{2}\), x=0, y=2, and y=5. Text Transcription: x 5y - y^2

Consider the region bounded by y=m x, y=0, x=0, and x=b. (a) Find the upper and lower sums to approximate the area of the region when \(\Delta x=b / 4\). (b) Find the upper and lower sums to approximate the area of the region when \(\Delta x=b / n\). (c) Find the area of the region by letting n approach infinity in both sums in part (b). Show that, in each case, you obtain the formula for the area of a triangle. Text Transcription: \Delta x=b / 4

In Exercises 27 and 28, sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral. \(\int_{0}^{5}(5-|x-5|) d x\) Text Transcription: \int_{0}^{5}(5-|x-5|) d x

In Exercises 27 and 28, sketch the region whose area is given by the definite integral. Then use a geometric formula to evaluate the integral. \(\int_{-6}^{6} \sqrt{36-x^{2}} d x\) Text Transcription: \int_{-6}^{6} \sqrt{36-x^{2} d x

In Exercises 31–36, use the Fundamental Theorem of Calculus to evaluate the definite integral. \(\int_{0}^{8}(3+x) d x\) Text Transcription: \int_{0}^{8}(3+x) d x

In Exercises 31–36, use the Fundamental Theorem of Calculus to evaluate the definite integral. \(\int_{2}^{3}\left(x^{4}+4 x-6\right) d x\) Text Transcription: \int_{2}^{3}(x^{4}+4 x-6) d x

In Exercises 31–36, use the Fundamental Theorem of Calculus to evaluate the definite integral. \(\int_{4}^{9} x \sqrt{x} d x\) Text Transcription: \int_{4}^{9} x \sqrt{x} d x

In Exercises 31–36, use the Fundamental Theorem of Calculus to evaluate the definite integral. \(\int_{-\pi / 4}^{\pi / 4} \sec ^{2} t d t\) Text Transcription: \int_{-\pi / 4}^{\pi / 4} \sec ^{2} t d t

In Exercises 31–36, use the Fundamental Theorem of Calculus to evaluate the definite integral. \(\int_{0}^{2}\left(x+e^{x}\right) d x\) Text Transcription: \int_{0}^{2}\left(x+e^{x}\right) d x

In Exercises 31–36, use the Fundamental Theorem of Calculus to evaluate the definite integral. \(\int_{1}^{6} \frac{3}{x} d x\) Text Transcription: \int_{1}^{6} \frac{3}{x} d x

In Exercises 37–40, find the area of the region bounded by the graphs of the equations. \(y=8-x, \quad x=0, \quad x=6, \quad y=0\) Text Transcription: y=8-x, \quad x=0, \quad x=6, \quad y=0

In Exercises 37–40, find the area of the region bounded by the graphs of the equations. \(y=\sqrt{x}(1-x), \quad y=0\) Text Transcription: y=\sqrt{x}(1-x), y=0

In Exercises 37–40, find the area of the region bounded by the graphs of the equations. \(y=\frac{2}{x}, \quad y=0, \quad x=1, \quad x=3\) Text Transcription: y=\frac{2}{x}, \quad y=0, \quad x=1, \quad x=3

In Exercises 37–40, find the area of the region bounded by the graphs of the equations. \(y=1+e^{x}, \quad y=0, \quad x=0, \quad x=2\) Text Transcription: y=1+e^{x}, \quad y=0, \quad x=0, \quad x=2

In Exercises 41 and 42, find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. \(f(x)=\frac{1}{\sqrt{x}}, \quad[4,9]\) Text Transcription: f(x)=\frac{1}{\sqrt{x}, \quad[4,9]

In Exercises 41 and 42, find the average value of the function over the given interval and all values of x in the interval for which the function equals its average value. \(f(x)=x^{3}, \quad[0,2]\) Text Transcription: f(x)=x^{3}, \quad[0,2]

In Exercises 43-46, use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). \(F(x)=\int_{0}^{x} t^{2} \sqrt{1+t^{3}} d t\) Text Transcription: F’(x) F(x)=\int_{0}^{x} t^{2} \sqrt{1+t^{3} d t

In Exercises 43-46, use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). \(F(x)=\int_{1}^{x} \frac{1}{t^{2}} d t\) Text Transcription: F’(x) F(x)=\int_{1}^{x} \frac{1}{t^{2} d t

In Exercises 43-46, use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). \(F(x)=\int_{-3}^{x}\left(t^{2}+3 t+2\right) d t\) Text Transcription: F’(x) F(x)=\int_{-3}^{x}(t^{2}+3 t+2) d t

In Exercises 43-46, use the Second Fundamental Theorem of Calculus to find \(F^{\prime}(x)\). \(F(x)=\int_{0}^{x} \csc ^{2} t d t\) Text Transcription: F’(x) F(x)=\int_{0}^{x} \csc ^{2} t d t

In Exercises 47–60, find the indefinite integral. \(\int \frac{x^{2}}{\sqrt{x^{3}+3}} d x\) Text Transcription: \int \frac x^2 \sqrt x^3+3 d x

In Exercises 47–60, find the indefinite integral. \(\int 6 x^{3} \sqrt{3 x^{4}+2} d x\) Text Transcription: \int 6 x^{3} \sqrt{3 x^{4}+2} d x

In Exercises 47–60, find the indefinite integral. \(\int x\left(1-3 x^{2}\right)^{4} d x\) Text Transcription: \int x\left(1-3 x^{2}\right)^{4} d x

In Exercises 47–60, find the indefinite integral. \(\int \frac{x+4}{\left(x^{2}+8 x-7\right)^{2}} d x\) Text Transcription: \int \frac x+4 (x^{2}+8 x-7)^2 d x

In Exercises 47–60, find the indefinite integral. \(\int \sin ^{3} x \cos x d x\) Text Transcription: \int \sin ^{3} x \cos x d x

In Exercises 47–60, find the indefinite integral. \(\int x \sin 3 x^{2} d x\) Text Transcription: \int x \sin 3 x^{2} d x

In Exercises 47–60, find the indefinite integral. \(\int \frac{\cos \theta}{\sqrt{1-\sin \theta}} d \theta\) Text Transcription: \int \frac \cos \theta \sqrt 1-\sin \theta d \theta

In Exercises 47–60, find the indefinite integral. \(\int \frac{\sin x}{\sqrt{\cos x}} d x\) Text Transcription: \int \frac{\sin x}{\sqrt{\cos x}} d x

In Exercises 47–60, find the indefinite integral. \(\int x e^{-3 x^{2}} d x\) Text Transcription: \int x e^{-3 x^{2}} d x

In Exercises 47–60, find the indefinite integral. \(\int \frac{e^{1 / x}}{x^{2}} d x\) Text Transcription: \int \frac{e^{1 / x}}{x^{2}} d x

In Exercises 47–60, find the indefinite integral. \(\int(x+1) 5^{(x+1)^{2}} d x\) Text Transcription: \int(x+1) 5^{(x+1)^{2}} d x

In Exercises 47–60, find the indefinite integral. \(\int \frac{1}{t^{2}}\left(2^{-1 / t}\right) d t\) Text Transcription: \int \frac1 t^{2(2^{-1 / t) d t

In Exercises 47–60, find the indefinite integral. \(\int(1+\sec \pi x)^{2} \sec \pi x \tan \pi x d x\) Text Transcription: \int(1+\sec \pi x)^{2} \sec \pi x \tan \pi x d x

In Exercises 47–60, find the indefinite integral. \(\int \sec 2 x \tan 2 x d x\) Text Transcription: \int \sec 2 x \tan 2 x d x

In Exercises 61–68, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{0}^{1}(3 x+1)^{5} d x\) Text Transcription: \int_{0}^{1}(3 x+1)^{5} d x

In Exercises 61–68, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{0}^{1} x^{2}\left(x^{3}-2\right)^{3} d x\) Text Transcription: \int_{0}^{1} x^{2}\left(x^{3}-2\right)^{3} d x

In Exercises 61–68, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{0}^{3} \frac{1}{\sqrt{1+x}} d x\) Text Transcription: \int_{0}^{3} \frac 1 \sqrt 1+x d x

In Exercises 61–68, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{3}^{6} \frac{x}{3 \sqrt{x^{2}-8}} d x\) Text Transcription: \int_{3}^{6} \frac{x}{3 \sqrt{x^{2}-8}} d x

In Exercises 61–68, evaluate the definite integral. Use a graphing utility to verify your result. \(2 \pi \int_{0}^{1}(y+1) \sqrt{1-y} d y\) Text Transcription: 2 \pi \int_{0}^{1}(y+1) \sqrt{1-y} d y

In Exercises 61–68, evaluate the definite integral. Use a graphing utility to verify your result. \(2 \pi \int_{-1}^{0} x^{2} \sqrt{x+1} d x\) Text Transcription: 2 \pi \int_{-1}^{0} x^{2} \sqrt{x+1} d x

In Exercises 61–68, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{0}^{\pi} \cos \frac{x}{2} d x\) Text Transcription: \int_{0}^{\pi} \cos \frac{x}{2} d x

In Exercises 61–68, evaluate the definite integral. Use a graphing utility to verify your result. \(\int_{-\pi / 4}^{\pi / 4} \sin 2 x d x\) Text Transcription: \int_{-\pi / 4}^{\pi / 4} \sin 2 x d x

In Exercises 69–72, approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n=4. Compare these results with the approximation of the integral using a graphing utility. \(\int_{2}^{3} \frac{2}{1+x^{2}} d x\) Text Transcription: \int_{2}^{3} \frac{2}{1+x^{2}} d x

In Exercises 69–72, approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n=4. Compare these results with the approximation of the integral using a graphing utility. \(\int_{0}^{1} \frac{x^{3 / 2}}{3-x^{2}} d x\) Text Transcription: \int_{0}^{1} \frac{x^{3 / 2}}{3-x^{2}} d x

In Exercises 69–72, approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n=4. Compare these results with the approximation of the integral using a graphing utility. \(\int_{0}^{3} \sqrt{x} \ln (x+1) d x\) Text Transcription: \int_{0}^{3} \sqrt{x} \ln (x+1) d x

In Exercises 69–72, approximate the definite integral using the Trapezoidal Rule and Simpson’s Rule with n=4. Compare these results with the approximation of the integral using a graphing utility. \(\int_{0}^{\pi} \sqrt{1+\sin ^{2} x} \ d x\) Text Transcription: int_{0}^{\pi} \sqrt{1+\sin ^{2} x} d x

In Exercises 73–78, find the indefinite integral. \(\int \frac{1}{7 x-2} d x\) Text Transcription: \int \frac{1}{7 x-2} d x

In Exercises 73–78, find the indefinite integral. \(\int \frac{x^{2}}{x^{3}+1} d x\) Text Transcription: \int \frac{x^{2}}{x^{3}+1} d x

In Exercises 73–78, find the indefinite integral. \(\int \frac{\sin x}{1+\cos x} d x\) Text Transcription: \int \frac{\sin x}{1+\cos x} d x

In Exercises 73–78, find the indefinite integral. \(\int \frac{\ln \sqrt{x}}{x} d x\) Text Transcription: \int \frac \ln \sqrtx x d x

In Exercises 73–78, find the indefinite integral. \(\int \frac{e^{2 x}-e^{-2 x}}{e^{2 x}+e^{-2 x}} d x\) Text Transcription: \int \frac e^2 x-e^-2 x e^2 x+e^-2 x d x

In Exercises 73–78, find the indefinite integral. \(\int \frac{e^{2 x}}{e^{2 x}+1} d x\) Text Transcription: \int \frac{e^{2 x}}{e^{2 x}+1} d x

In Exercises 79–82, evaluate the definite integral. \(\int_{1}^{4} \frac{2 x+1}{2 x} d x\) Text Transcription: \int_{1}^{4} \frac{2 x+1}{2 x} d x

In Exercises 79–82, evaluate the definite integral. \(\int_{1}^{e} \frac{\ln x}{x} d x\) Text Transcription: \int_{1}^{e} \frac{\ln x}{x} d x

In Exercises 79–82, evaluate the definite integral. \(\int_{0}^{\pi / 3} \sec \theta d \theta\) Text Transcription: \int_{0}^{\pi / 3} \sec \theta d \theta

In Exercises 79–82, evaluate the definite integral. \(\int_{0}^{\pi} \tan \frac{\theta}{3} d \theta\) Text Transcription: \int_{0}^{\pi} \tan \frac{\theta}{3} d \theta

In Exercises 83–88, find the indefinite integral. \(\int \frac{1}{e^{2 x}+e^{-2 x}} d x\) Text Transcription: \int \frac 1 e^2 x+e^-2 x d x

In Exercises 83–88, find the indefinite integral. \(\int \frac{1}{3+25 x^{2}} d x\) Text Transcription: \int \frac{1}{3+25 x^{2}} d x

In Exercises 83–88, find the indefinite integral. \(\int \frac{x}{\sqrt{1-x^{4}}} d x\) Text Transcription: \int \frac{x}{\sqrt{1-x^{4}}} d x

In Exercises 83–88, find the indefinite integral. \(\int \frac{1}{x \sqrt{9 x^{2}-49}} d x\) Text Transcription: \int \frac 1 x \sqrt{9 x^2-49 d x

In Exercises 83–88, find the indefinite integral. \(\int \frac{\arctan (x / 2)}{4+x^{2}} d x\) Text Transcription: \int \frac \arctan (x / 2)4+x^{2} d x

In Exercises 83–88, find the indefinite integral. \(\int \frac{\arcsin 2 x}{\sqrt{1-4 x^{2}}} d x\) Text Transcription: \int \frac \arcsin 2 x \sqrt 1-4 x^2 d x

In Exercises 89–92, find the derivative of the function. \(y=\operatorname{sech}(4 x-1)\) Text Transcription: y=sech(4 x-1)

In Exercises 89–92, find the derivative of the function. \(y=2 x-\cosh \sqrt{x}\) Text Transcription: y=2 x-\cosh \sqrt{x}

In Exercises 89–92, find the derivative of the function. \(y=\sinh ^{-1}(4 x)\) Text Transcription: y=\sinh ^{-1}(4 x)

In Exercises 89–92, find the derivative of the function. \(y=x \tanh ^{-1} 2 x\) Text Transcription: y=x \tanh ^{-1} 2 x

In Exercises 93–96, find the indefinite integral. \(\int x^{2} \operatorname{sech}^{2} x^{3} d x\) Text Transcription: \int x^2 sech{2 x^3 d x

In Exercises 93–96, find the indefinite integral. \(\int \sinh 6 x d x\) Text Transcription: \int \sinh 6 x d x

In Exercises 93–96, find the indefinite integral. \(\int \frac{1}{9-4 x^{2}} d x\) Text Transcription: \int \frac{1}{9-4 x^{2}} d x

In Exercises 93–96, find the indefinite integral. \(\int \frac{x}{\sqrt{x^{4}-1}} d x\) Text Transcription: \int \frac{x}{\sqrt{x^{4}-1}} d x

Given \(\int_{4}^{8} f(x) d x=12\) and \(\int_{4}^{8} g(x) d x=5\) evaluate (a) \(\int_{4}^{8}[f(x)+g(x)] d x\). (b) \(\int_{4}^{8}[f(x)-g(x)] d x\). (c) \(\int_{4}^{8}[2 f(x)-3 g(x)] d x\). (d) \(\int_{4}^{8} 7 f(x) d x\). Text Transcription: \int_{4}^{8} f(x) d x=1 \int_{4}^{8} g(x) d x=5 \int_{4}^{8}[f(x)+g(x)] d \int_{4}^{8}[f(x)-g(x)] d x \int_{4}^{8}[2 f(x)-3 g(x)] d x \int_{4}^{8} 7 f(x) d x

Given \(\int_{0}^{3} f(x) d x=4\) and \(\int_{3}^{6} f(x) d x=-1\) evaluate (a) \(\int_{0}^{6} f(x) d x\). (b) \(\int_{6}^{3} f(x) d x\). (c) \(\int_{4}^{4} f(x) d x\). (d) \(\int_{3}^{6}-10 f(x) d x\). Text Transcription: int_{0}^{3} f(x) d x=4 int_{3}^{6} f(x) d x=-1 int_{0}^{6} f(x) d x. int_{6}^{3} f(x) d x. int_{4}^{4} f(x) d x. int_{3}^{6}-10 f(x) d x