QuestionEvaluate the integral $\int x e^{-x^{2}} \, dx$ .

Studdy Solution

STEP 1

Assumptions1. We are given the integral $\int x e^{-x^{}} d x$ . We will use the method of integration by substitution to solve this integral.

STEP 2

Choose the substitution $u = x^2$ . This is a good choice because the derivative of $u$ with respect to $x$ is $2x$ , which is present in the integral.
$u = x^2$

STEP 3

Calculate the derivative of $u$ with respect to $x$ .
$\frac{du}{dx} =2x$

STEP 4

Rearrange the derivative to express $dx$ in terms of $du$ .
$dx = \frac{du}{2x}$

STEP 5

Substitute $u$ and $dx$ into the integral.
$\int x e^{-u} \frac{du}{2x}$

STEP 6

implify the integral by cancelling out the $x$ terms.
$\frac{1}{2} \int e^{-u} du$

STEP 7

The integral of $e^{-u}$ with respect to $u$ is $-e^{-u}$ .
$\frac{1}{2} \int e^{-u} du = -\frac{1}{2} e^{-u} + C$

STEP 8

Substitute $u = x^2$ back into the integral to get the final answer.
$-\frac{1}{2} e^{-x^2} + C$ So, the integral $\int x e^{-x^{2}} d x$ equals $-\frac{1}{2} e^{-x^2} + C$ .

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QuestionEvaluate the integral $\int x e^{-x^{2}} \, dx$ .

Studdy Solution

STEP 1

Assumptions1. We are given the integral $\int x e^{-x^{}} d x$ . We will use the method of integration by substitution to solve this integral.

STEP 2

Choose the substitution $u = x^2$ . This is a good choice because the derivative of $u$ with respect to $x$ is $2x$ , which is present in the integral.
$u = x^2$

STEP 3

Calculate the derivative of $u$ with respect to $x$ .
$\frac{du}{dx} =2x$

STEP 4

Rearrange the derivative to express $dx$ in terms of $du$ .
$dx = \frac{du}{2x}$

STEP 5

Substitute $u$ and $dx$ into the integral.
$\int x e^{-u} \frac{du}{2x}$

STEP 6

implify the integral by cancelling out the $x$ terms.
$\frac{1}{2} \int e^{-u} du$

STEP 7

The integral of $e^{-u}$ with respect to $u$ is $-e^{-u}$ .
$\frac{1}{2} \int e^{-u} du = -\frac{1}{2} e^{-u} + C$

STEP 8

Substitute $u = x^2$ back into the integral to get the final answer.
$-\frac{1}{2} e^{-x^2} + C$ So, the integral $\int x e^{-x^{2}} d x$ equals $-\frac{1}{2} e^{-x^2} + C$ .

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QuestionEvaluate the integral ∫xe−x2 dx\int x e^{-x^{2}} \, dx∫xe−x2dx.

Studdy Solution

STEP 1

Assumptions1. We are given the integral ∫xe−xdx\int x e^{-x^{}} d x∫xe−xdx . We will use the method of integration by substitution to solve this integral.

STEP 2

Choose the substitution u=x2u = x^2u=x2. This is a good choice because the derivative of uuu with respect to xxx is 2x2x2x, which is present in the integral.u=x2u = x^2u=x2

STEP 3

Calculate the derivative of uuu with respect to xxx.dudx=2x\frac{du}{dx} =2xdxdu​=2x

STEP 4

Rearrange the derivative to express dxdxdx in terms of dududu.dx=du2xdx = \frac{du}{2x}dx=2xdu​

STEP 5

Substitute uuu and dxdxdx into the integral.∫xe−udu2x\int x e^{-u} \frac{du}{2x}∫xe−u2xdu​

STEP 6

implify the integral by cancelling out the xxx terms.12∫e−udu\frac{1}{2} \int e^{-u} du21​∫e−udu

STEP 7

The integral of e−ue^{-u}e−u with respect to uuu is −e−u-e^{-u}−e−u.12∫e−udu=−12e−u+C\frac{1}{2} \int e^{-u} du = -\frac{1}{2} e^{-u} + C21​∫e−udu=−21​e−u+C

STEP 8

Substitute u=x2u = x^2u=x2 back into the integral to get the final answer.−12e−x2+C-\frac{1}{2} e^{-x^2} + C−21​e−x2+CSo, the integral ∫xe−x2dx\int x e^{-x^{2}} d x∫xe−x2dx equals −12e−x2+C-\frac{1}{2} e^{-x^2} + C−21​e−x2+C.

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QuestionEvaluate the integral $\int x e^{-x^{2}} \, dx$ .

Assumptions1. We are given the integral $\int x e^{-x^{}} d x$ . We will use the method of integration by substitution to solve this integral.

Choose the substitution $u = x^2$ . This is a good choice because the derivative of $u$ with respect to $x$ is $2x$ , which is present in the integral.
$u = x^2$

Calculate the derivative of $u$ with respect to $x$ .
$\frac{du}{dx} =2x$

Rearrange the derivative to express $dx$ in terms of $du$ .
$dx = \frac{du}{2x}$

Substitute $u$ and $dx$ into the integral.
$\int x e^{-u} \frac{du}{2x}$

implify the integral by cancelling out the $x$ terms.
$\frac{1}{2} \int e^{-u} du$

The integral of $e^{-u}$ with respect to $u$ is $-e^{-u}$ .
$\frac{1}{2} \int e^{-u} du = -\frac{1}{2} e^{-u} + C$

Substitute $u = x^2$ back into the integral to get the final answer.
$-\frac{1}{2} e^{-x^2} + C$ So, the integral $\int x e^{-x^{2}} d x$ equals $-\frac{1}{2} e^{-x^2} + C$ .