?In the following exercises, use a calculator to graph the antiderivative \(\int f\)

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. \(\int_{0}^{\sqrt{3} / 2} \frac{d x}{\sqrt{1-x^{2}}}\) Text Transcription: int_0^sqrt 3/2 dx/sqrt 1-x^2

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. \(\int_{-1 / 2}^{1 / 2} \frac{d x}{\sqrt{1-x^{2}}}\) Text Transcription: int_-1/2^1/2 dx/sqrt 1-x^2

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. \(\int_{\sqrt{3}}^{1} \frac{d x}{\sqrt{1+x^{2}}}\) Text Transcription: int_sqrt 3^1 dx/sqrt 1+x^2

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. \(\int_{1 / \sqrt{3}}^{\sqrt{3}} \frac{d x}{1+x^{2}}\) Text Transcription: int_1/sqrt 3^sqrt 3 dx/1+x^2

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. \(\int_{1}^{\sqrt{2}} \frac{d x}{|x| \sqrt{x^{2}-1}}\) Text Transcription: \int_1^sqrt 2 dx/|x| sqrt x^2-1

In the following exercises, evaluate each integral in terms of an inverse trigonometric function. \(\int_{1}^{2 / \sqrt{3}} \frac{d x}{|x| \sqrt{x^{2}-1}}\) Text Transcription: int_1^2/sqrt 3 dx/|x| sqrt x^2-1

In the following exercises, find each indefinite integral, using appropriate substitutions. \(\int \frac{d x}{\sqrt{9-x^{2}}}\) Text Transcription: int dx/sqrt 9-x^2

In the following exercises, find each indefinite integral, using appropriate substitutions. \(\int \frac{d x}{25+16 x^{2}}\) Text Transcription: int dx/25+16x^2

In the following exercises, find each indefinite integral, using appropriate substitutions. \(\int \frac{d x}{|x| \sqrt{x^{2}-9}}\) Text Transcription: int dx/|x| sqrt x^2-9

In the following exercises, find each indefinite integral, using appropriate substitutions. \(\int \frac{d x}{|x| \sqrt{4 x^{2}-16}}\) Text Transcription: int dx/|x| sqrt 4x^2-16

Explain the relationship \(-\cos ^{-1} t+C=\int \frac{d t}{\sqrt{1-t^{2}}}=\sin ^{-1} t+C\). Is it true, in general, that \(\cos ^{-1} t=-\sin ^{-1} t\)? Text Transcription: -cos^-1 t+C=int dt/sqrt 1-t^2=sin^-1 t+C cos^-1 t=-sin^-1 t

Explain the relationship \(\sec ^{-1} t+C=\int \frac{d t}{|t| \sqrt{t^{2}-1}}=-\csc ^{-1} t+C\). Is it true, in general, that \(\sec ^{-1} t=-\csc ^{-1} t\)? Text Transcription: sec^-1 t+C=int dt/|t| sqrt t^2-1=-csc^-1 t+C\) sec^-1 t=-csc^-1 t

In the following exercises, find each indefinite integral, using appropriate substitutions. \(\int \frac{d x}{\sqrt{1-16 x^{2}}}\) Text Transcription: int dx/sqrt 1-16x^2

In the following exercises, find each indefinite integral, using appropriate substitutions. \(\int \frac{d x}{9+x^{2}}\) Text Transcription: int dx/9+x^2

Explain what is wrong with the following integral: \(\int_{1}^{2} \frac{d t}{\sqrt{1-t^{2}}}\) Text Transcription: int_1^2 dt/sqrt 1-t^2

Explain what is wrong with the following integral: \(\int_{-1}^{1} \frac{d t}{|t| \sqrt{t^{2}-1}}\) Text Transcription: int_-1^1 dt/|t| sqrt t^2-1

In the following exercises, solve for the antiderivative \(\int f\) of f with C=0, then use a calculator to graph f and the antiderivative over the given interval [a, b]. Identify a value of C such that adding C to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{1}{\sqrt{9-x^{2}}} d x\) over [-3,3] Text Transcription: int f F(x)=int_a^x f(t) dt int 1/sqrt 9-x^2 dx

In the following exercises, solve for the antiderivative \(\int f\) of f with C=0, then use a calculator to graph f and the antiderivative over the given interval [a, b]. Identify a value of C such that adding C to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{9}{9+x^{2}} d x\) over [-6,6] Text Transcription: int f F(x)=int_a^x f(t) dt int 9/9+x^2 dx

In the following exercises, solve for the antiderivative \(\int f\) of f with C=0, then use a calculator to graph f and the antiderivative over the given interval [a, b]. Identify a value of C such that adding C to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{\cos x}{4+\sin ^{2} x} d x\) over [-6,6] Text Transcription: int f F(x)=int_a^x f(t) dt int cos x/4+sin^2 x dx

In the following exercises, solve for the antiderivative \(\int f\) of f with C=0, then use a calculator to graph f and the antiderivative over the given interval [a, b]. Identify a value of C such that adding C to the antiderivative recovers the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{e^{x}}{1+e^{2 x}} d x\) over [-6,6] Text Transcription: int f F(x)=int_a^x f(t) dt int e^x/1+e^2x dx

In the following exercises, compute the antiderivative using appropriate substitutions. \(\int \frac{\sin ^{-1} t d t}{\sqrt{1-t^{2}}}\) Text Transcription: int sin^-1 tdt/sqrt 1-t^2

In the following exercises, compute the antiderivative using appropriate substitutions. \(\int \frac{d t}{\sin ^{-1} t \sqrt{1-t^{2}}}\) Text Transcription: int dt/sin^-1 t sqrt 1-t^2

In the following exercises, compute the antiderivative using appropriate substitutions. \(\int \frac{\tan ^{-1}(2 t)}{1+4 t^{2}} d t\) Text Transcription: int tan^-1(2 t)/1+4 t^2 dt

In the following exercises, compute the antiderivative using appropriate substitutions. \(\int \frac{t \tan ^{-1}\left(t^{2}\right)}{1+t^{4}} d t\) Text Transcription: int t tan^-1(t^2)/1+t^4 dt

In the following exercises, compute the antiderivative using appropriate substitutions. \(\int \frac{\sec ^{-1}\left(\frac{t}{2}\right)}{|t| \sqrt{t^{2}-4}} d t\) Text Transcription: int sec^-1(t/2)/|t| sqrt t^2-4 dt

In the following exercises, compute the antiderivative using appropriate substitutions. \(\int \frac{t \sec ^{-1}\left(t^{2}\right)}{t^{2} \sqrt{t^{4}-1}} d t\) Text Transcription: int t sec^-1(t^2)/t^2 sqrt t^4-1 dt

In the following exercises, use a calculator to graph the antiderivative \(\int f\) with C=0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{1}{x \sqrt{x^{2}-4}} d x\) over [2,6] Text Transcription: int f F(x)=int_a^x f(t) dt int 1/x sqrt x^2 -4

In the following exercises, use a calculator to graph the antiderivative \(\int f\) with C=0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{1}{(2 x+2) \sqrt{x}} d x\) over [0,6] Text Transcription: int f F(x)=int_a^x f(t) dt int 1/(2x+2) sqrt x dx

In the following exercises, use a calculator to graph the antiderivative \(\int f\) with C=0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{(\sin x+x \cos x)}{1+x^{2} \sin ^{2} x} d x\) over [-6,6] Text Transcription: int f F(x)=int_a^x f(t) dt int (sin x+x cos x)/1+x^2 sin^2 x dx

In the following exercises, use a calculator to graph the antiderivative \(\int f\) with C=0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{2 e^{-2 x}}{\sqrt{1-e^{-4 x}}} d x\) over [0,2] Text Transcription: int f F(x)=int_a^x f(t) dt int 2e^-2x/sqrt 1-e^-4 x dx

In the following exercises, use a calculator to graph the antiderivative \(\int f\) with C=0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{1}{x+x \ln ^{2} x}\) over [0,2] Text Transcription: int f F(x)=int_a^x f(t) dt int 1/x+x ln^2 x

In the following exercises, use a calculator to graph the antiderivative \(\int f\) with C=0 over the given interval [a, b]. Approximate a value of C, if possible, such that adding C to the antiderivative gives the same value as the definite integral \(F(x)=\int_{a}^{x} f(t) d t\). [T] \(\int \frac{\sin ^{-1} x}{\sqrt{1-x^{2}}}\) over [-1,1] Text Transcription: int f F(x)=int_a^x f(t) dt int sin^-1 x/ sqrt 1-x^2

In the following exercises, compute each integral using appropriate substitutions. \(\int \frac{e^{x}}{\sqrt{1-e^{2 t}}} d t\) Text Transcription: int e^x/sqrt 1-e^2t dt

In the following exercises, compute each integral using appropriate substitutions. \(\int \frac{e^{t}}{1+e^{2 t}} d t\) Text Transcription: int e^t/1+e^2t dt

In the following exercises, compute each integral using appropriate substitutions. \(\int \frac{d t}{t \sqrt{1-\ln ^{2} t}}\) Text Transcription: int dt/t sqrt 1-ln^2 t

In the following exercises, compute each integral using appropriate substitutions. \(\int \frac{d t}{t\left(1+\ln ^{2} t\right)}\) Text Transcription: int dt/t(1+ln^2 t)

In the following exercises, compute each integral using appropriate substitutions. \(\int \frac{\cos ^{-1}(2 t)}{\sqrt{1-4 t^{2}}} d t\) Text Transcription: int cos^-1(2 t)/sqrt 1-4 t^2 dt

In the following exercises, compute each integral using appropriate substitutions. \(\int \frac{e^{t} \cos ^{-1}\left(e^{t}\right)}{\sqrt{1-e^{2 t}}} d t\) Text Transcription: int e^t cos^-1(e^t)/sqrt 1-e^2 t dt

In the following exercises, compute each definite integral. \(\int_{0}^{1 / 2} \frac{\tan \left(\sin ^{-1} t\right)}{\sqrt{1-t^{2}}} d t\) Text Transcription: int_0^1/2 tan (sin^-1 t)/sqrt 1-t^2 dt

In the following exercises, compute each definite integral. \(\int_{1 / 4}^{1 / 2} \frac{\tan \left(\cos ^{-1} t\right)}{\sqrt{1-t^{2}}} d t\) Text Transcription: int_1/4^1/2 tan(cos^-1 t)/sqrt 1-t^2 dt

In the following exercises, compute each definite integral. \(\int_{0}^{1 / 2} \frac{\sin \left(\tan ^{-1} t\right)}{1+t^{2}} d t\) Text Transcription: int_0^1/2 sin(tan^-1 t)/1+t^2 dt

In the following exercises, compute each definite integral. \(\int_{0}^{1 / 2} \frac{\cos \left(\tan ^{-1} t\right)}{1+t^{2}} d t\) Text Transcription: int_0^1/2 cos (tan^-1 t)/1+t^2 dt

For A>0, compute \(I(A)=\int_{-A}^{A} \frac{d t}{1+t^{2}}\) and evaluate \(\lim _{a \rightarrow \infty} I(A)\), the area under the graph of \(\frac{1}{1+t^{2}}\) on \([-\infty, \infty]\). Text Transcription: I(A)=int_-A^A dt/1+t^2 lim_a rightarrow infty I(A) 1/1+t^2 [-infty, infty]

For \(1<B<\infty\), compute \(I(B)=\int_{1}^{B} \frac{d t}{t \sqrt{t^{2}-1}}\) and evaluate \(\lim _{B \rightarrow \infty} I(B)\), the area under the graph of \(\frac{1}{t \sqrt{t^{2}-1}}\) over \([1, \infty)\). Text Transcription: 1<B<\infty I(B)=int_1^B dt/t sqrt t^2-1 lim_B rightarrow infty I(B 1/t sqrt t^2 -1 [1, infty]

Use the substitution \(u=\sqrt{2} \cot x\) and the identity \(1+\cot ^{2} x=\csc ^{2} x\) to evaluate \(\int \frac{d x}{1+\cos ^{2} x}\). (Hint: Multiply the top and bottom of the integrand by \(\csc ^{2} x\). Text Transcription: u=sqrt 2 cot 1+cot^2 x=csc^2 x int dx/1+cos^2 x csc^2 x

[T] Approximate the points at which the graphs of \(f(x)=2 x^{2}-1\) and \(g(x)=\left(1+4 x^{2}\right)^{-3 / 2}\) intersect, and approximate the area between their graphs accurate to three decimal places. Text Transcription: f(x)=2x^2-1 g(x)=(1+4x^2)^-3/2

[T] Approximate the points at which the graphs of \(f(x)=x^{2}-1\) and \(f(x)=x^{2}-1\) intersect, and approximate the area between their graphs accurate to three decimal places. Text Transcription: f(x)=x^2-1

Use the following graph to prove that \(\int_{0}^{x} \sqrt{1-t^{2}} d t=\frac{1}{2} x \sqrt{1-x^{2}}+\frac{1}{2} \sin ^{-1} x\) Text Transcription: int_0^x sqrt 1-t^2 dt=1/2 x sqrt 1-x^2+1/2 sin^-1 x