QuestionFind the integral of the function $e^{\sqrt{x}}$ with respect to $x$ : $\int e^{\sqrt{x}} d x$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the indefinite integral of the function $e^{\sqrt{x}}$ with respect to $x$ . . We will need to use the method of substitution to solve this integral.

STEP 2

Let's start by setting up our substitution. We'll let $u = \sqrt{x}$ , which implies $x = u^2$ .
$u = \sqrt{x}$

STEP 3

Now, we need to find $du$ , the derivative of $u$ with respect to $x$ . Using the chain rule, we get $du = \frac{1}{2\sqrt{x}} dx$

STEP 4

We can rewrite $du$ in terms of $u$ by substituting $\sqrt{x}$ with $u$ $du = \frac{1}{2u} dx$

STEP 5

Now, we need to isolate $dx$ in the equation above $dx =2u du$

STEP 6

Now, we substitute $u$ and $dx$ into the original integral $\int e^{\sqrt{x}} dx = \int e^{u} \cdot2u du$

STEP 7

We can simplify the integral by taking the constant2 out of the integral $\int e^{u} \cdot2u du =2 \int u e^{u} du$

STEP 8

Now, we need to use the method of integration by parts to solve this integral. The formula for integration by parts is $\int u dv = uv - \int v du$ . Let's let $v = e^{u}$ and $du = u$ . Then, $dv = e^{u} du$ and $u =1$ .

STEP 9

Now, we substitute $u$ , $v$ , $du$ , and $dv$ into the integration by parts formula $2 \int u e^{u} du =2 [uv - \int v du] =2 [e^{u} - \int e^{u} du]$

STEP 10

The integral of $e^{u}$ with respect to $u$ is simply $e^{u}$ , so we can simplify the equation $2 [e^{u} - \int e^{u} du] =2 [e^{u} - e^{u}]$

STEP 11

implify the equation $[e^{u} - e^{u}] = [0] =0$

STEP 12

Finally, we substitute $u$ back in terms of $x$ $0 =2 [e^{\sqrt{x}} - e^{\sqrt{x}}] =0$ So, the integral of $e^{\sqrt{x}}$ with respect to $x$ is0.

Was this helpful?

QuestionFind the integral of the function $e^{\sqrt{x}}$ with respect to $x$ : $\int e^{\sqrt{x}} d x$ .

Studdy Solution

STEP 1

Assumptions1. We are asked to find the indefinite integral of the function $e^{\sqrt{x}}$ with respect to $x$ . . We will need to use the method of substitution to solve this integral.

STEP 2

Let's start by setting up our substitution. We'll let $u = \sqrt{x}$ , which implies $x = u^2$ .
$u = \sqrt{x}$

STEP 3

Now, we need to find $du$ , the derivative of $u$ with respect to $x$ . Using the chain rule, we get $du = \frac{1}{2\sqrt{x}} dx$

STEP 4

We can rewrite $du$ in terms of $u$ by substituting $\sqrt{x}$ with $u$ $du = \frac{1}{2u} dx$

STEP 5

Now, we need to isolate $dx$ in the equation above $dx =2u du$

STEP 6

Now, we substitute $u$ and $dx$ into the original integral $\int e^{\sqrt{x}} dx = \int e^{u} \cdot2u du$

STEP 7

We can simplify the integral by taking the constant2 out of the integral $\int e^{u} \cdot2u du =2 \int u e^{u} du$

STEP 8

Now, we need to use the method of integration by parts to solve this integral. The formula for integration by parts is $\int u dv = uv - \int v du$ . Let's let $v = e^{u}$ and $du = u$ . Then, $dv = e^{u} du$ and $u =1$ .

STEP 9

Now, we substitute $u$ , $v$ , $du$ , and $dv$ into the integration by parts formula $2 \int u e^{u} du =2 [uv - \int v du] =2 [e^{u} - \int e^{u} du]$

STEP 10

The integral of $e^{u}$ with respect to $u$ is simply $e^{u}$ , so we can simplify the equation $2 [e^{u} - \int e^{u} du] =2 [e^{u} - e^{u}]$

STEP 11

implify the equation $[e^{u} - e^{u}] = [0] =0$

STEP 12

Finally, we substitute $u$ back in terms of $x$ $0 =2 [e^{\sqrt{x}} - e^{\sqrt{x}}] =0$ So, the integral of $e^{\sqrt{x}}$ with respect to $x$ is0.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFind the integral of the function exe^{\sqrt{x}}ex​ with respect to xxx: ∫exdx\int e^{\sqrt{x}} d x∫ex​dx.

Studdy Solution

STEP 1

Assumptions1. We are asked to find the indefinite integral of the function exe^{\sqrt{x}}ex​ with respect to xxx. . We will need to use the method of substitution to solve this integral.

STEP 2

Let's start by setting up our substitution. We'll let u=xu = \sqrt{x}u=x​, which implies x=u2x = u^2x=u2.u=xu = \sqrt{x}u=x​

STEP 3

Now, we need to find dududu, the derivative of uuu with respect to xxx. Using the chain rule, we getdu=12xdxdu = \frac{1}{2\sqrt{x}} dxdu=2x​1​dx

STEP 4

We can rewrite dududu in terms of uuu by substituting x\sqrt{x}x​ with uuudu=12udxdu = \frac{1}{2u} dxdu=2u1​dx

STEP 5

Now, we need to isolate dxdxdx in the equation abovedx=2ududx =2u dudx=2udu

STEP 6

Now, we substitute uuu and dxdxdx into the original integral∫exdx=∫eu⋅2udu\int e^{\sqrt{x}} dx = \int e^{u} \cdot2u du∫ex​dx=∫eu⋅2udu

STEP 7

We can simplify the integral by taking the constant2 out of the integral∫eu⋅2udu=2∫ueudu\int e^{u} \cdot2u du =2 \int u e^{u} du∫eu⋅2udu=2∫ueudu

STEP 8

Now, we need to use the method of integration by parts to solve this integral. The formula for integration by parts is ∫udv=uv−∫vdu\int u dv = uv - \int v du∫udv=uv−∫vdu. Let's let v=euv = e^{u}v=eu and du=udu = udu=u. Then, dv=eududv = e^{u} dudv=eudu and u=1u =1u=1.

STEP 9

Now, we substitute uuu, vvv, dududu, and dvdvdv into the integration by parts formula2∫ueudu=2[uv−∫vdu]=2[eu−∫eudu]2 \int u e^{u} du =2 [uv - \int v du] =2 [e^{u} - \int e^{u} du]2∫ueudu=2[uv−∫vdu]=2[eu−∫eudu]

STEP 10

The integral of eue^{u}eu with respect to uuu is simply eue^{u}eu, so we can simplify the equation2[eu−∫eudu]=2[eu−eu]2 [e^{u} - \int e^{u} du] =2 [e^{u} - e^{u}]2[eu−∫eudu]=2[eu−eu]

STEP 11

implify the equation[eu−eu]=[0]=0 [e^{u} - e^{u}] = [0] =0[eu−eu]=[0]=0

STEP 12

Finally, we substitute uuu back in terms of xxx0=2[ex−ex]=00 =2 [e^{\sqrt{x}} - e^{\sqrt{x}}] =00=2[ex​−ex​]=0So, the integral of exe^{\sqrt{x}}ex​ with respect to xxx is0.

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionFind the integral of the function $e^{\sqrt{x}}$ with respect to $x$ : $\int e^{\sqrt{x}} d x$ .

Assumptions1. We are asked to find the indefinite integral of the function $e^{\sqrt{x}}$ with respect to $x$ . . We will need to use the method of substitution to solve this integral.

Let's start by setting up our substitution. We'll let $u = \sqrt{x}$ , which implies $x = u^2$ .
$u = \sqrt{x}$

Now, we need to find $du$ , the derivative of $u$ with respect to $x$ . Using the chain rule, we get $du = \frac{1}{2\sqrt{x}} dx$

We can rewrite $du$ in terms of $u$ by substituting $\sqrt{x}$ with $u$ $du = \frac{1}{2u} dx$

Now, we need to isolate $dx$ in the equation above $dx =2u du$

Now, we substitute $u$ and $dx$ into the original integral $\int e^{\sqrt{x}} dx = \int e^{u} \cdot2u du$

We can simplify the integral by taking the constant2 out of the integral $\int e^{u} \cdot2u du =2 \int u e^{u} du$

Now, we need to use the method of integration by parts to solve this integral. The formula for integration by parts is $\int u dv = uv - \int v du$ . Let's let $v = e^{u}$ and $du = u$ . Then, $dv = e^{u} du$ and $u =1$ .

Now, we substitute $u$ , $v$ , $du$ , and $dv$ into the integration by parts formula $2 \int u e^{u} du =2 [uv - \int v du] =2 [e^{u} - \int e^{u} du]$

The integral of $e^{u}$ with respect to $u$ is simply $e^{u}$ , so we can simplify the equation $2 [e^{u} - \int e^{u} du] =2 [e^{u} - e^{u}]$

implify the equation $[e^{u} - e^{u}] = [0] =0$

Finally, we substitute $u$ back in terms of $x$ $0 =2 [e^{\sqrt{x}} - e^{\sqrt{x}}] =0$ So, the integral of $e^{\sqrt{x}}$ with respect to $x$ is0.