Solved: ?Evaluate the indefinite integral. Illustrate, and check that your answer is

Evaluate the integral. \(\int \sin ^{3} x \cos ^{2} x d x\) Equation Transcription: ? Text Transcription: integral sin^3 x cos^2 x dx

Evaluate the integral. \(\int \cos ^{6} y \sin ^{3} y d y\) Equation Transcription: ? Text Transcription: integral cos^6 y sin^3 y dy

Evaluate the integral. \(\int_{0}^{\pi / 2} \cos ^{9} x \sin ^{5} x d x\) Equation Transcription: Text Transcription: integral _0 ^pi/2 cos^9 x sin^5 x dx

Evaluate the integral. \(\int_{0}^{\pi / 4} \sin ^{5} x d x\) Equation Transcription: Text Transcription: integral _0 ^pi/4 sin^5 x dx

Evaluate the integral. \(\int \sin ^{5}(2 t) \cos ^{2}(2 t) d t\) Equation Transcription: ? Text Transcription: integral sin^5 (2t) cos^2 (2t) dt

Evaluate the integral. \(\int \cos ^{3}(t / 2) \sin ^{2}(t / 2) d t\) Equation Transcription: ? Text Transcription: integral cos^3 (t/2) sin^2 (t/2) dt

Evaluate the integral. \(\int_{0}^{\pi / 2} \cos ^{2} \theta \ d \theta\) Equation Transcription: Text Transcription: integral _0 ^pi/2 cos^2 theta d theta

Evaluate the integral. \(\int_{0}^{\pi / 4} \sin ^{2}(2 \theta) \ d \theta\)

Evaluate the integral. \(\int_{0}^{\pi} \cos ^{4}(2 t) d t\) Equation Transcription: Text Transcription: integral _0 ^pi cos^4 (2t) dt

Evaluate the integral. \(\int_{0}^{\pi} \sin ^{2} t \cos ^{4} t \ d t\) Equation Transcription: Text Transcription: integral _0 ^pi sin^2 t cos^4 t dt

Evaluate the integral. \(\int_{0}^{\pi / 2} \sin ^{2} x \cos ^{2} x \ d x\) Equation Transcription: ? Text Transcription: integral _0 ^pi/2 sin^2 x cos^2 x dx

Evaluate the integral. \(\int_{0}^{\pi / 2}(2-\sin \theta)^{2} d \theta\)

Evaluate the integral. \(\int \sqrt{\cos \theta} \sin ^{3} \theta d \theta\) Equation Transcription: ? Text Transcription: integral sqrt cos theta sin^3 theta d theta

Evaluate the integral. \(\int(1+\sqrt[3]{\sin t}) \cos ^{3} t d t\) Equation Transcription: ? Text Transcription: integral (1+ cube root sin t) cos^3 t dt

Evaluate the integral. \(\int \sin x \sec ^{5} x d x\) Equation Transcription: ? Text Transcription: integral sin x sec^5 x dx

Evaluate the integral. \(\int \csc ^{5} \theta \cos ^{3} \theta d \theta\) Equation Transcription: ? Text Transcription: integral csc^5 theta cos^3 theta d theta

Evaluate the integral. \(\int \cot x \cos ^{2} x d x\)

Evaluate the integral. \(\int \tan ^{2} x \cos ^{3} x d x\) Equation Transcription: ? Text Transcription: integral _tan^2 x cos^3 x dx

Evaluate the integral. \(\int \sin ^{2} x \sin 2 x d x\) Equation Transcription: ? Text Transcription: integral sin^2 x sin 2x dx

Evaluate the integral. \(\int \sin x \cos \left(\frac{1}{2} x\right) d x\) Equation Transcription: ? Text Transcription: integral sin x cos (1/2 x) dx

Evaluate the integral. \(\int \tan x \sec ^{3} x d x\) Equation Transcription: ? Text Transcription: integral tan x sec^3 x dx

Evaluate the integral. \(\int \tan ^{2} \theta \sec ^{4} \theta d \theta\) Equation Transcription: ? Text Transcription: integral tan^2 theta sec^4 theta d theta

Evaluate the integral. \(\int \tan ^{2} x d x\)

Evaluate the integral. \(\int\left(\tan ^{2} x+\tan ^{4} x\right) d x\) Equation Transcription: ? Text Transcription: integral (tan^2 x + tan^4 x) dx

Evaluate the integral. \(\int \tan ^{4} x \sec ^{6} x d x\) Equation Transcription: ? Text Transcription: integral tan^4 x sec^6 x dx

Evaluate the integral. \(\int_{0}^{\pi / 4} \sec ^{6} \theta \tan ^{6} \theta d \theta\) Equation Transcription: Text Transcription: integral _0 ^pi/4 sec^6 theta tan^6 theta d theta

Evaluate the integral. \(\int \tan ^{3} x \sec x d x\) Equation Transcription: ? Text Transcription: integral tan^3 x sec x dx

Evaluate the integral. \(\int \tan ^{5} x \sec ^{3} x d x\)

Evaluate the integral. \(\int \tan ^{3} x \sec ^{6} x d x\)

Evaluate the integral. \(\int_{0}^{\pi / 4} \tan ^{4} t d t\) Equation Transcription: Text Transcription: integral _0 ^pi/4 tan^4 t dt

Evaluate the integral. \(\int \tan ^{5} x d x\)

Evaluate the integral. \(\int \tan ^{2} x \sec x d x\) Equation Transcription: ? Text Transcription: integral tan^2 x sec x dx

Evaluate the integral. \(\int \frac{1-\tan ^{2} x}{\sec ^{2} x} d x\) Equation Transcription: Text Transcription: integral 1-tan^2 x/sec^2 x dx

Evaluate the integral. \(\int_{ }^{ }\frac{\tan x\sec^2x}{\cos x}dx\) Equation Transcription: Text Transcription: integral tan x sec^2/cos x dx

Evaluate the integral. \(\int_{0}^{\pi / 4} \frac{\sin ^{3} x}{\cos x} d x\) Equation Transcription: Text Transcription: Integral 0 pi/4 sin^3x/cos x dx

Evaluate the integral. \(\int \frac{\sin \theta+\tan \theta}{\cos ^{3} \theta} d \theta\) Equation Transcription: Text Transcription: the integral fraction with numerator sin theta plus tangent theta and denominator theta power cosine d theta

Evaluate the integral. \(\int_{\pi / 6}^{\pi / 2} \cot ^{2} x \ d x\) Equation Transcription: Text Transcription: integral_pi/6 ^pi/2 cot^2 x dx

Evaluate the integral. \(\int_{\pi / 4}^{\pi / 2} \cot ^{3} x d x\)

Evaluate the integral. \(\int_{\pi / 4}^{\pi / 2} \cot ^{5}\phi c s c^{3} \phi d \phi\) Equation Transcription: ???? ???? ???? Text Transcription: integral pi/4 pi/2 cot^5 phi csc^3 phi d phi

Evaluate the integral. \(\int_{\pi / 4}^{\pi / 2} \csc ^{4} \theta \cot ^{4} \theta d \theta\) Equation transcription: Text Transcription: Integral pi/4 pi/2 csc^4 omega d omega

Evaluate the integral. \(\int \csc x d x\)

Evaluate the integral. \(\int_{\pi / 6}^{\pi / 3} \csc ^{3} x d x\) Equation Transcription: Text Transcription: Integral pi/6 pi/3 csc^3 x dx

Evaluate the integral. \(\int \sin 8 x \cos 5 x d x\) Equation Transcription: Text Transcription: integral sin 8x cos 5x dx

Evaluate the integral. \( \int \sin 2 \theta \sin 6 \theta d \theta\) Equation Transcription: Text Transcription: integral sin 2 omega sin 6 omega d omega

Evaluate the integral. \(\int_{0}^{\pi / 2} \cos 5 t \cos 10 t d t\)

Evaluate the integral. \(\int t \cos ^{5}\left(t^{2}\right) d t\)

Evaluate the integral. \(\int \frac{\sin ^{2}(1 / t)}{t^{2}} d t\)

Evaluate the integral. \(\int \frac{\sin ^{2}(1 / t)}{t^{2}} d t\) Equation transcription: Text Transcription: Integral sec^2y cos^3 (tan y) dy

Evaluate the integral. \(\int_{0}^{\pi / 6} \sqrt{1+\cos 2 x} d x\) Equation Transcription: Text Transcription: Integral 0 pi/6 square root 1 + cos 2x dx

Evaluate the integral. \(\int_{0}^{\pi / 4} \sqrt{1-\cos 4 \theta} d \theta\)

Evaluate the integral. \(\int t \sin ^{2} t d t\) Equation Transcription: Text Transcription: integral t sin^2 t dt

Evaluate the integral. \(\int x \sec x \tan x d x\) Equation Transcription: Text Transcription: Integral x sec x tan x dx

Evaluate the integral. \(\int x \tan ^{2} x d x\) Equation Transcription: Text Transcription: Integral x tan^2x dx

Evaluate the integral. \(\int x \sin ^{3} x d x \mid\)

Evaluate the integral. \(\int \frac{d x}{\cos x-1}\) Equation Transcription: Text Transcription: integral dx/cos x-1

Evaluate the integral. \(\int \frac{1}{\sec \theta+1} d \theta\) Equation Transcription: Text Transcription: Integral 1/sec omega + 1 d omega

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking \( (C= 0)\)). \(\int x \sin ^{2}\left(x^{2}\right) d x\) Equation Transcription: Text Transcription: Integral x sin^2(x^2)dx c=0

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking . \(\int \sin 3 x \sin 6 x d x\) Equation Transcription: Text Transcription: Integral sin 3x sin 6x dx

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking \(C=0\)). \(\int \sin 3 x \sin 6 x d x\) Equation Transcription: Text Transcription: Integral sin 3x sin 6x dx

Evaluate the indefinite integral. Illustrate, and check that your answer is reasonable, by graphing both the integrand and its antiderivative (taking . \(\int \sec ^{4}\left(\frac{1}{2} x\right) d x\) Equation Transcription: Text Transcription: Integral sec^4 (1/2x)dx

If \(\int_{0}^{\pi / 4} \tan ^{6} x \sec x d x=I\), express the value of \(\int_{0}^{\pi / 4} \tan ^{8} x \sec x d x\) in terms of . Equation Transcription: , Text Transcription: Integral 0 pi/4 tan^6 sec x dx = I , integral 0 pi/4 tan^8 x sec x dx

(a) Prove the reduction formula \(\int \tan ^{2 n} x d x=\frac{\tan ^{2 n-1} x}{2 n-1}-\int \tan ^{2 n-2} x d x\) (b) Use this formula to find \(\int \tan ^{8} x d x\) Equation Transcription: Text Transcription: Integral tan^2n x dx = tan^2n-1/2n-1 - integral tan^2n-2x dx Integral tan^8x dx

Find the average value of the function \(f(x)=\sin ^{2} x \cos ^{3} x\) on the interval \([-\pi, \pi]\).

Evaluate \(\int \sin x \cos x d x\) by four methods: (a) the substitution \(u=\cos x\) (b) the substitution \(u=\sin x\) (c) the identity \(\sin 2 x=2 \sin x \cos x\) (d) integration by parts. Explain the different appearances of the answers. Equation Transcription: Text Transcription: Integral sin x cos x dx U = cos x U = sin x Sin 2x = 2 sin x cos x

Find the area of the region bounded by the given curves. \(y=\sin ^{2} x, \quad y=\sin ^{3} x, \quad 0 \leqslant x \leqslant \pi\) Equation Transcription: 0 ? x ? Image Text Transcription: Y = sin^2x, y = sin^3x, 0 leqslant x leqslant pi

Find the area of the region bounded by the given curves. \(y=\tan x, y=\tan ^{2} x, 0 \leqslant x \leqslant \pi / 4\) Equation Transcription: 0 ? x ? Text Transcription: Y = yan x, y = tan^2x, 0 ? x ? pi/4

Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct. \(\int_{0}^{2 \pi} \cos ^{3} x d x\) Equation Transcript: Text Transcription: Integral 0 2pi cos^3 x dx

Use a graph of the integrand to guess the value of the integral. Then use the methods of this section to prove that your guess is correct. \(\int_{0}^{2} \sin 2 \pi x \cos 5 \pi x d x\) Equation Transcription: Text Transcription: Integral 0 2 2 pi x cos 5 pi x dx

Find the volume obtained by rotating the region bounded by the curves about the given axis. \(y=\sin x, \quad y=0, \quad \pi / 2 \leqslant x \leqslant \pi\) about the -axis Equation Transcription: ? x ? Text Transcription: Y = sin x, y = 0, pi/2 leqslant x leqslant pi

Find the volume obtained by rotating the region bounded by the curves about the given axis. \(y=\sin ^{2} x, y=0,0 \leqslant x<\pi\) about the -axis Equation Transcription: 0 ? x ? Text Transcription: Y = sin^2x, y = 0, 0 leqslant x leqslant pi;

Find the volume obtained by rotating the region bounded by the curves about the given axis. \(y=\sin x, \quad y=\cos x, \quad 0 \leqslant x \leqslant \pi / 4\) about Equation Transcription: 0 ? x ? Text Transcription: Y = sin x, y = 0, 0 leqslant x leqslant pi/4;

Find the volume obtained by rotating the region bounded by the curves about the given axis. \(y=\sec x, y=\cos x, 0 \leqslant x \leqslant \pi / 3\) about y=-1

A particle moves on a straight line with velocity function \(v(t)=\sin\omega t\cos^2\omega t\). Find its position function \(s=f(t) \text { if } f(0)=0\). Equation transcription: Text Transcription: v(t) = sin omega t cos^2 omega t s= f (t) f(0) = 0

Household electricity is supplied in the form of alternating current that varies from to with a frequency of 60 cycles per second . The voltage is thus given by the equation \(E(t)=155 \sin (120 \pi t)\) where is the time in seconds. Voltmeters read the RMS (root-mean-square) voltage, which is the square root of the average value of over one cycle. (a) Calculate the RMS voltage of household current. (b) Many electric stoves require an RMS voltage of . Find the corresponding amplitude needed for the voltage \(E(t)=A \sin (120 \pi t)\). Equation transcription: Text Transcription: E (t) = 155 sin(120 pi t) E(t) = A sin(120 pi t) Bracket E(t) bracket^2

Prove the formula, where and are positive integers. \(\int_{-\pi}^{\pi} \sin m x \cos n x d x=0\) Equation Transcription: Text Transcription: Integral -pi pi sin nx cos nx dx = 0

Prove the formula, where and are positive integers.

Prove the formula, where \(m\) and \(n\) are positive integers. \(\int_{-\pi}^{\pi} \cos m x \cos n x d x= \begin{cases}0 & \text { if } m \neq n \\ \pi & \text { if } m=n\end{cases}\)

A finite Fourier series is given by the sum \(f(x)=\sum_{n=1}^{N} a_{n} \sin n x\) \(=a_{1} \sin x+a_{2} \sin 2 x+\cdots+a_{N} \sin N x\) Use the result of Exercise 76 to show that the coefficient is given by the formula \(a_{m}=\frac{1}{\pi} \int_{-\pi}^{\pi} f(x) \sin m x d x\) Equation Transcription: ???+ Text Transcription: f(x) = subscript n=1 N a_n sin nx =a_1 sin x + a_x sin 2x + dot dot dot + a_N sin Nx A_m = 1/pi integral -pi pi f(x) sin mx dx