적분법 전체 문제

Why is u-substitution referred to as change of variable?

2. If $f=g\circ h,$ when reversing the chain rule, $\frac{d}{dx}(g\circ h)(x)={g}^{'}(h(x)){h}^{'}(x),$ should you take $u=g(x)$ or $u=h(x)?$

In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $\int f(u)du.$

$\int x\sqrt{x+1}dx=\frac{2}{15}{(x+1)}^{3/2}(3x-2)+C;u=x+1$

In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $\int f(u)du.$

For $x>1:\int \frac{{x}^{2}}{\sqrt{x-1}}dx=\frac{2}{15}\sqrt{x-1}(3{x}^{2}+4x+8)+C;u=x-1$

In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $\int f(u)du.$

$\int x\sqrt{4{x}^{2}+9}dx=\frac{1}{12}{(4{x}^{2}+9)}^{3/2}+C;u=4{x}^{2}+9$

In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $\int f(u)du.$

$\int \frac{x}{\sqrt{4{x}^{2}+9}}dx=\frac{1}{4}\sqrt{4{x}^{2}+9}+C;u=4{x}^{2}+9$

In the following exercises, verify each identity using differentiation. Then, using the indicated u-substitution, identify f such that the integral takes the form $\int f(u)du.$

$\int \frac{x}{{(4{x}^{2}+9)}^{2}}dx=-\frac{1}{8(4{x}^{2}+9)};u=4{x}^{2}+9$

In the following exercises, find the antiderivative using the indicated substitution.

$\int {(x+1)}^{4}dx;u=x+1$

In the following exercises, find the antiderivative using the indicated substitution.

$\int {(x-1)}^{5}dx;u=x-1$

In the following exercises, find the antiderivative using the indicated substitution.

$\int {(2x-3)}^{-7}dx;u=2x-3$

In the following exercises, find the antiderivative using the indicated substitution.

$\int {(3x-2)}^{-11}dx;u=3x-2$

In the following exercises, find the antiderivative using the indicated substitution.

$\int \frac{x}{\sqrt{1-{x}^{2}}}dx;u=1-{x}^{2}$

In the following exercises, find the antiderivative using the indicated substitution.

$\int (x-1){({x}^{2}-2x)}^{3}dx;u={x}^{2}-2x$

In the following exercises, find the antiderivative using the indicated substitution.

$\int ({x}^{2}-2x){({x}^{3}-3{x}^{2})}^{2}dx;u={x}^{3}-3{x}^{2}$

In the following exercises, find the antiderivative using the indicated substitution.

$\int {\cos }^{3}\theta d\theta ;u=\sin \theta$ $(Hint:\,{\cos }^{2}\theta =1-{\sin }^{2}\theta )$

In the following exercises, find the antiderivative using the indicated substitution.

$\int {\sin }^{3}\theta d\theta ;u=\cos \theta$ $(Hint:\,{\sin }^{2}\theta =1-{\cos }^{2}\theta )$

In the following exercises, use a suitable change of variables to determine the indefinite integral.

$\int x{(1-x)}^{99}dx$

In the following exercises, use a suitable change of variables to determine the indefinite integral.

$\int t{(1-{t}^{2})}^{10}dt$

In the following exercises, use a suitable change of variables to determine the indefinite integral.

$\int {(11x-7)}^{-3}dx$

In the following exercises, use a suitable change of variables to determine the indefinite integral.

$\int {(7x-11)}^{4}dx$