Chapter 5

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int(11 x-7)^{-3} d x $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int(7 x-11)^{4} d x $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \cos ^{3} \theta \sin \theta d \theta $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \sin ^{7} \theta \cos \theta d \theta $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \cos ^{2}(\pi t) \sin (\pi t) d t $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \sin ^{2} x \cos ^{3} x d x \quad\left(\operatorname{Hint} \sin ^{2} x+\cos ^{2} x=1\right) $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int t \sin \left(t^{2}\right) \cos \left(t^{2}\right) d t $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int t^{2} \cos ^{2}\left(t^{3}\right) \sin \left(t^{3}\right) d t $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \frac{x^{2}}{\left(x^{3}-3\right)^{2}} d x $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \frac{x^{3}}{\sqrt{1-x^{2}}} d x $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \frac{y^{5}}{\left(1-y^{3}\right)^{3 / 2}} d y $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int \cos \theta(1-\cos \theta)^{99} \sin \theta d \theta $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int\left(1-\cos ^{3} \theta\right)^{10} \cos ^{2} \theta \sin \theta d \theta $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int(\cos \theta-1)\left(\cos ^{2} \theta-2 \cos \theta\right)^{3} \sin \theta d \theta $$

In the following exercises, use a suitable change of variables to determine the indefinite integral. $$ \int\left(\sin ^{2} \theta-2 \sin \theta\right)\left(\sin ^{3} \theta-3 \sin ^{2} \theta\right)^{3} \cos \theta d \theta $$

In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer. $$ y=3(1-x)^{2} \text { over }[0,2] $$

In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer. $$ y=x\left(1-x^{2}\right)^{3} \text { over }[-1,2] $$

In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer. $$ y=\sin x(1-\cos x)^{2} \text { over }[0, \pi] $$

In the following exercises, use a calculator to estimate the area under the curve using left Riemann sums with 50 terms, then use substitution to solve for the exact answer. $$ y=\frac{x}{\left(x^{2}+1\right)^{2}} \text { over }[-1,1] $$

In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{1} x \sqrt{1-x^{2}} d x $$

In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{1} \frac{x}{\sqrt{1+x^{2}}} d x $$

In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{2} \frac{t}{\sqrt{5+t^{2}}} d t $$

In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{1} \frac{t}{\sqrt{1+t^{3}}} d t $$

In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{\pi / 4} \sec ^{2} \theta \tan \theta d \theta $$

In the following exercises, use a change of variables to evaluate the definite integral. $$ \int_{0}^{\pi / 4} \frac{\sin \theta}{\cos ^{4} \theta} d \theta $$

In the following exercises, evaluate the indefinite integral \(\int f(x) d x\) with constant \(C=0\) using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of \(C\) that would need to be added to the antiderivative to make it equal to the definite integral \(F(x)=\int_{a}^{x} f(t) d t,\) with a the left endpoint of the given interval. $$ \int(2 x+1) e^{x^{2}+x-6} d x \text { over }[-3,2] $$

In the following exercises, evaluate the indefinite integral \(\int f(x) d x\) with constant \(C=0\) using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of \(C\) that would need to be added to the antiderivative to make it equal to the definite integral \(F(x)=\int_{a}^{x} f(t) d t,\) with a the left endpoint of the given interval. $$ \int \frac{\cos (\ln (2 x))}{x} d x \text { on }[0,2] $$

In the following exercises, evaluate the indefinite integral \(\int f(x) d x\) with constant \(C=0\) using \(u\) -substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of \(C\) that would need to be added to the antiderivative to make it equal to the definite integral \(F(x)=\int_{a}^{x} f(t) d t,\) with \(a\) the left endpoint of the given interval. 298\. [T] \(\int(2 x+1) e^{x^{2}+x-6} d x\) over [-3,2] 299\. [T] \(\int \frac{\cos (\ln (2 x))}{x} d x\) on [0,2] 300\. [T] \(\int \frac{3 x^{2}+2 x+1}{\sqrt{x^{3}+x^{2}+x+4}} d x\) over [-1,2] 301\. [T] \(\int \frac{\sin x}{\cos ^{3} x} d x\) over \(\left[-\frac{\pi}{3}, \frac{\pi}{3}\right]\) 302\. [T] \(\int(x+2) e^{-x^{2}-4 x+3} d x\) over [-5,1] 303\. [T] \(\int 3 x^{2} \sqrt{2 x^{3}+1} d x\) over [0,1]

In the following exercises, evaluate the indefinite integral \(\int f(x) d x\) with constant \(C=0\) using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of \(C\) that would need to be added to the antiderivative to make it equal to the definite integral \(F(x)=\int_{a}^{x} f(t) d t,\) with a the left endpoint of the given interval. $$ \int \frac{\sin x}{\cos ^{3} x} d x \text { over }\left[-\frac{\pi}{3}, \frac{\pi}{3}\right] $$

In the following exercises, evaluate the indefinite integral \(\int f(x) d x\) with constant \(C=0\) using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of \(C\) that would need to be added to the antiderivative to make it equal to the definite integral \(F(x)=\int_{a}^{x} f(t) d t,\) with a the left endpoint of the given interval. $$ \int(x+2) e^{-x^{2}-4 x+3} d x \text { over }[-5,1] $$

In the following exercises, evaluate the indefinite integral \(\int f(x) d x\) with constant \(C=0\) using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of \(C\) that would need to be added to the antiderivative to make it equal to the definite integral \(F(x)=\int_{a}^{x} f(t) d t,\) with a the left endpoint of the given interval. $$ \int 3 x^{2} \sqrt{2 x^{3}+1} d x \text { over }[0,1] $$

In the following exercises, evaluate the indefinite integral \(\int f(x) d x\) with constant \(C=0\) using u-substitution. Then, graph the function and the antiderivative over the indicated interval. If possible, estimate a value of \(C\) that would need to be added to the antiderivative to make it equal to the definite integral \(F(x)=\int_{a}^{x} f(t) d t,\) with a the left endpoint of the given interval. If \(h(a)=h(b)\) in \(\int_{a}^{b} g^{\prime} h(x) h_{h}(x) d x,\) what can you say about the value of the integral?

Is the substitution \(u=1-x^{2}\) in the definite integral \(\int_{0}^{2} \frac{x}{1-x^{2}} d x\) okay? If not, why not?

In the following exercises, use a change of variables to show that each definite integral is equal to zero. $$ \int_{0}^{\pi} \cos ^{2}(2 \theta) \sin (2 \theta) d \theta $$

In the following exercises, use a change of variables to show that each definite integral is equal to zero. $$ \int_{0}^{\sqrt{\pi}} t \cos \left(t^{2}\right) \sin \left(t^{2}\right) d t $$

In the following exercises, use a change of variables to show that each definite integral is equal to zero. $$ \int_{0}^{1}(1-2 t) d t $$

In the following exercises, use a change of variables to show that each definite integral is equal to zero. \(\int_{0}^{1} \frac{1-2 t}{\left(1+\left(t-\frac{1}{2}\right)^{2}\right)} d t\)

In the following exercises, use a change of variables to show that each definite integral is equal to zero. $$ \int_{0}^{\pi} \sin \left(\left(t-\frac{\pi}{2}\right)^{3}\right) \cos \left(t-\frac{\pi}{2}\right) d t $$

In the following exercises, use a change of variables to show that each definite integral is equal to zero. $$ \int_{0}^{2}(1-t) \cos (\pi t) d t $$

In the following exercises, use a change of variables to show that each definite integral is equal to zero. $$ \int_{\pi / 4}^{3 \pi / 4} \sin ^{2} t \cos t d t $$

Show that the average value of \(f(x)\) over an interval \([a, b]\) is the same as the average value of \(f(c x)\) over the interval \(\left[\frac{a}{c}, \frac{b}{c}\right]\) for \(c>0\)

Find the area under the graph of \(f(t)=\frac{t}{\left(1+t^{2}\right)^{a}}\) between \(t=0\) and \(t=x\) where \(a>0\) and \(a \neq 1\) is fixed, and evaluate the limit as \(x \rightarrow \infty\) .

Find the area under the graph of \(g(t)=\frac{t}{\left(1-t^{2}\right)^{a}}\) between \(t=0\) and \(t=x,\) where \(0 < x < 1\) and \(a > 0\) is fixed. Evaluate the limit as \(x \rightarrow 1\)

The area of a semicircle of radius 1 can be expressed as \(\int_{-1}^{1} \sqrt{1-x^{2}} d x .\) Use the substitution \(x=\cos t\) to express the area of a semicircle as the integral of a trigonometric function. You do not need to compute the integral.

The area of the top half of an ellipse with a major axis that is the \(x\) -axis from \(x=-1\) to \(a\) and with a minor axis that is the \(y\) -axis from \(y=-b\) to \(b\) can be written as \(\int_{-a}^{a} b \sqrt{1-\frac{x^{2}}{a^{2}}} d x .\) Use the substitution \(x=a \cos t\) to express this area in terms of an integral of a trigonometric function. You do not need to compute the integral.

In the following exercises, compute each indefinite integral. $$\int e^{2 x} d x$$

In the following exercises, compute each indefinite integral. $$\int e^{-3 x} d x$$

In the following exercises, compute each indefinite integral. $$\int 2^{x} d x$$

In the following exercises, compute each indefinite integral. $$\int 3^{-x} d x$$

In the following exercises, compute each indefinite integral. $$\int \frac{1}{2 x} d x$$