적분법 전체 문제

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, evaluate the indefinite integral $\int f(x)dx$ 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 $F(x)={\int }_{a}^{x}f(t)dt,$ with a the left endpoint of the given interval.

[T] $\int (2x+1){e}^{{x}^{2}+x-6}dx$ over $[-3,2]$

In the following exercises, evaluate the indefinite integral $\int f(x)dx$ 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 $F(x)={\int }_{a}^{x}f(t)dt,$ with a the left endpoint of the given interval.

[T] $\int \frac{\cos (\ln (2x))}{x}dx$ on $[0,2]$

In the following exercises, evaluate the indefinite integral $\int f(x)dx$ 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 $F(x)={\int }_{a}^{x}f(t)dt,$ with a the left endpoint of the given interval.

[T] $\int \frac{3{x}^{2}+2x+1}{\sqrt{{x}^{3}+{x}^{2}+x+4}}dx$ over $[-1,2]$

In the following exercises, evaluate the indefinite integral $\int f(x)dx$ 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 $F(x)={\int }_{a}^{x}f(t)dt,$ with a the left endpoint of the given interval.

[T] $\int \frac{\sin x}{{\cos }^{3}x}dx$ over $\left[-\frac{\pi }{3},\frac{\pi }{3}\right]$

In the following exercises, evaluate the indefinite integral $\int f(x)dx$ 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 $F(x)={\int }_{a}^{x}f(t)dt,$ with a the left endpoint of the given interval.

[T] $\int (x+2){e}^{-{x}^{2}-4x+3}dx$ over $[-5,1]$

In the following exercises, evaluate the indefinite integral $\int f(x)dx$ 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 $F(x)={\int }_{a}^{x}f(t)dt,$ with a the left endpoint of the given interval.

[T] $\int 3{x}^{2}\sqrt{2{x}^{3}+1}dx$ over $[0,1]$

If $h(a)=h(b)$ in ${\int }_{a}^{b}g'(h(x))h'(x)dx,$ what can you say about the value of the integral?

Is the substitution $u=1-{x}^{2}$ in the definite integral ${\int }_{0}^{2}\frac{x}{1-{x}^{2}}dx$ 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.

${\int }_{0}^{\sqrt{\pi }}t\cos ({t}^{2})\sin ({t}^{2})dt$

In the following exercises, use a change of variables to show that each definite integral is equal to zero.

${\int }_{0}^{1}(1-2t)dt$

In the following exercises, use a change of variables to show that each definite integral is equal to zero.

${\int }_{0}^{1}\frac{1-2t}{\left(1+{\left(t-\frac{1}{2}\right)}^{2}\right)}dt$

In the following exercises, use a change of variables to show that each definite integral is equal to zero.

${\int }_{0}^{\pi }\sin \left({\left(t-\frac{\pi }{2}\right)}^{3}\right)\cos \left(t-\frac{\pi }{2}\right)dt$

In the following exercises, use a change of variables to show that each definite integral is equal to zero.

${\int }_{0}^{2}(1-t)\cos (\pi t)dt$

In the following exercises, use a change of variables to show that each definite integral is equal to zero.

${\int }_{\pi /4}^{3\pi /4}{\sin }^{2}t\cos tdt$

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

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

The area of a semicircle of radius 1 can be expressed as ${\int }_{-1}^{1}\sqrt{1-{x}^{2}}dx.$ 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 $x=-a$ to $x=a$ and with a minor axis that is the y-axis from $y=-b$ to $y=b$ can be written as ${\int }_{-a}^{a}b\sqrt{1-\frac{{x}^{2}}{{a}^{2}}}dx.$ 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.