Math / Calculus

QuestionProblem 2. (1 point) Evaluate the following integral. Use substitution if necessary. Remember to write your answer in terms of the original variable $(x)$ . $\int x\left(8+x^{2}\right)^{5} d x=\frac{1}{12}\left(8+x^{2}\right)^{6}+C$

Studdy Solution

STEP 1

1. The integral is of the form $\int x(8 + x^2)^5 \, dx$ .
2. We will use substitution to simplify the integration process.
3. The answer must be expressed in terms of the original variable $x$ .

STEP 2

1. Identify a suitable substitution.
2. Perform the substitution.
3. Integrate with respect to the new variable.
4. Substitute back to the original variable $x$ .
5. Verify the integration result.

STEP 3

Identify a suitable substitution. Notice that the expression $8 + x^2$ is inside the power, and its derivative $2x$ is present (up to a constant factor) in the integrand. This suggests the substitution:
Let $u = 8 + x^2$ .
Then, differentiate $u$ with respect to $x$ :
$\frac{du}{dx} = 2x$
This implies:
$du = 2x \, dx$
So, we can express $x \, dx$ in terms of $du$ :
$x \, dx = \frac{1}{2} du$

STEP 4

Perform the substitution in the integral:
$\int x(8 + x^2)^5 \, dx = \int (8 + x^2)^5 \cdot x \, dx$
Substitute $u = 8 + x^2$ and $x \, dx = \frac{1}{2} du$ :
$= \int u^5 \cdot \frac{1}{2} \, du$
$= \frac{1}{2} \int u^5 \, du$

STEP 5

Integrate with respect to the new variable $u$ :
$\frac{1}{2} \int u^5 \, du = \frac{1}{2} \cdot \frac{u^{6}}{6} + C$
$= \frac{1}{12} u^6 + C$

STEP 6

Substitute back to the original variable $x$ :
Since $u = 8 + x^2$ , we have:
$\frac{1}{12} u^6 + C = \frac{1}{12} (8 + x^2)^6 + C$

STEP 7

Verify the integration result by differentiating $\frac{1}{12} (8 + x^2)^6 + C$ with respect to $x$ and checking if it equals the original integrand $x(8 + x^2)^5$ .
The differentiation process should confirm the correctness of the integration.
The evaluated integral is:
$\int x(8 + x^2)^5 \, dx = \frac{1}{12} (8 + x^2)^6 + C$

Was this helpful?

QuestionProblem 2. (1 point) Evaluate the following integral. Use substitution if necessary. Remember to write your answer in terms of the original variable $(x)$ . $\int x\left(8+x^{2}\right)^{5} d x=\frac{1}{12}\left(8+x^{2}\right)^{6}+C$

Studdy Solution

STEP 1

1. The integral is of the form $\int x(8 + x^2)^5 \, dx$ .
2. We will use substitution to simplify the integration process.
3. The answer must be expressed in terms of the original variable $x$ .

STEP 2

1. Identify a suitable substitution.
2. Perform the substitution.
3. Integrate with respect to the new variable.
4. Substitute back to the original variable $x$ .
5. Verify the integration result.

STEP 3

STEP 4

Perform the substitution in the integral:
$\int x(8 + x^2)^5 \, dx = \int (8 + x^2)^5 \cdot x \, dx$
Substitute $u = 8 + x^2$ and $x \, dx = \frac{1}{2} du$ :
$= \int u^5 \cdot \frac{1}{2} \, du$
$= \frac{1}{2} \int u^5 \, du$

STEP 5

Integrate with respect to the new variable $u$ :
$\frac{1}{2} \int u^5 \, du = \frac{1}{2} \cdot \frac{u^{6}}{6} + C$
$= \frac{1}{12} u^6 + C$

STEP 6

Substitute back to the original variable $x$ :
Since $u = 8 + x^2$ , we have:
$\frac{1}{12} u^6 + C = \frac{1}{12} (8 + x^2)^6 + C$

STEP 7

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.

QuestionProblem 2. (1 point) Evaluate the following integral. Use substitution if necessary. Remember to write your answer in terms of the original variable (x)(x)(x). ∫x(8+x2)5dx=112(8+x2)6+C\int x\left(8+x^{2}\right)^{5} d x=\frac{1}{12}\left(8+x^{2}\right)^{6}+C∫x(8+x2)5dx=121​(8+x2)6+C

Studdy Solution

STEP 1

1. The integral is of the form ∫x(8+x2)5 dx\int x(8 + x^2)^5 \, dx∫x(8+x2)5dx.2. We will use substitution to simplify the integration process.3. The answer must be expressed in terms of the original variable xxx.

STEP 2

1. Identify a suitable substitution.2. Perform the substitution.3. Integrate with respect to the new variable.4. Substitute back to the original variable xxx.5. Verify the integration result.

STEP 3

STEP 4

STEP 5

Integrate with respect to the new variable u u u:12∫u5 du=12⋅u66+C \frac{1}{2} \int u^5 \, du = \frac{1}{2} \cdot \frac{u^{6}}{6} + C 21​∫u5du=21​⋅6u6​+C=112u6+C = \frac{1}{12} u^6 + C =121​u6+C

STEP 6

Substitute back to the original variable x x x:Since u=8+x2 u = 8 + x^2 u=8+x2, we have:112u6+C=112(8+x2)6+C \frac{1}{12} u^6 + C = \frac{1}{12} (8 + x^2)^6 + C 121​u6+C=121​(8+x2)6+C

STEP 7

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.

QuestionProblem 2. (1 point) Evaluate the following integral. Use substitution if necessary. Remember to write your answer in terms of the original variable $(x)$ . $\int x\left(8+x^{2}\right)^{5} d x=\frac{1}{12}\left(8+x^{2}\right)^{6}+C$

1. The integral is of the form $\int x(8 + x^2)^5 \, dx$ .
2. We will use substitution to simplify the integration process.
3. The answer must be expressed in terms of the original variable $x$ .

1. Identify a suitable substitution.
2. Perform the substitution.
3. Integrate with respect to the new variable.
4. Substitute back to the original variable $x$ .
5. Verify the integration result.

Integrate with respect to the new variable $u$ :
$\frac{1}{2} \int u^5 \, du = \frac{1}{2} \cdot \frac{u^{6}}{6} + C$
$= \frac{1}{12} u^6 + C$

Substitute back to the original variable $x$ :
Since $u = 8 + x^2$ , we have:
$\frac{1}{12} u^6 + C = \frac{1}{12} (8 + x^2)^6 + C$