Math / Calculus

QuestionMake the given substitution to evaluate the indefinite integral. $\begin{array}{l} \int 7(7 x+14)^{4} d x, u=7 x+14 \\ \int 7(7 x+14)^{4} d x=\square \end{array}$

Studdy Solution

STEP 1

1. We need to evaluate the indefinite integral using the given substitution.
2. The substitution $u = 7x + 14$ will simplify the integration process.
3. We will use the method of substitution to solve the integral.

STEP 2

1. Perform the substitution $u = 7x + 14$ .
2. Express $dx$ in terms of $du$ .
3. Substitute into the integral.
4. Integrate with respect to $u$ .
5. Substitute back to express the result in terms of $x$ .

STEP 3

Perform the substitution $u = 7x + 14$ .
Calculate the derivative of $u$ with respect to $x$ :
$\frac{du}{dx} = 7$
Thus, $dx = \frac{du}{7}$ .

STEP 4

Substitute $u = 7x + 14$ and $dx = \frac{du}{7}$ into the integral:
$\int 7(7x+14)^4 \, dx = \int 7u^4 \cdot \frac{du}{7}$
The $7$ terms cancel out:
$\int u^4 \, du$

STEP 5

Integrate with respect to $u$ :
$\int u^4 \, du = \frac{u^5}{5} + C$
where $C$ is the constant of integration.

STEP 6

Substitute back $u = 7x + 14$ to express the result in terms of $x$ :
$\frac{(7x + 14)^5}{5} + C$
Thus, the evaluated indefinite integral is:
$\boxed{\frac{(7x + 14)^5}{5} + C}$

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QuestionMake the given substitution to evaluate the indefinite integral. $\begin{array}{l} \int 7(7 x+14)^{4} d x, u=7 x+14 \\ \int 7(7 x+14)^{4} d x=\square \end{array}$

Studdy Solution

STEP 1

1. We need to evaluate the indefinite integral using the given substitution.
2. The substitution $u = 7x + 14$ will simplify the integration process.
3. We will use the method of substitution to solve the integral.

STEP 2

1. Perform the substitution $u = 7x + 14$ .
2. Express $dx$ in terms of $du$ .
3. Substitute into the integral.
4. Integrate with respect to $u$ .
5. Substitute back to express the result in terms of $x$ .

STEP 3

Perform the substitution $u = 7x + 14$ .
Calculate the derivative of $u$ with respect to $x$ :
$\frac{du}{dx} = 7$
Thus, $dx = \frac{du}{7}$ .

STEP 4

Substitute $u = 7x + 14$ and $dx = \frac{du}{7}$ into the integral:
$\int 7(7x+14)^4 \, dx = \int 7u^4 \cdot \frac{du}{7}$
The $7$ terms cancel out:
$\int u^4 \, du$

STEP 5

Integrate with respect to $u$ :
$\int u^4 \, du = \frac{u^5}{5} + C$
where $C$ is the constant of integration.

STEP 6

Substitute back $u = 7x + 14$ to express the result in terms of $x$ :
$\frac{(7x + 14)^5}{5} + C$
Thus, the evaluated indefinite integral is:
$\boxed{\frac{(7x + 14)^5}{5} + C}$

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QuestionMake the given substitution to evaluate the indefinite integral. ∫7(7x+14)4dx,u=7x+14∫7(7x+14)4dx=□\begin{array}{l} \int 7(7 x+14)^{4} d x, u=7 x+14 \\ \int 7(7 x+14)^{4} d x=\square \end{array}∫7(7x+14)4dx,u=7x+14∫7(7x+14)4dx=□​

Studdy Solution

STEP 1

1. We need to evaluate the indefinite integral using the given substitution.2. The substitution u=7x+14 u = 7x + 14 u=7x+14 will simplify the integration process.3. We will use the method of substitution to solve the integral.

STEP 2

1. Perform the substitution u=7x+14 u = 7x + 14 u=7x+14.2. Express dx dx dx in terms of du du du.3. Substitute into the integral.4. Integrate with respect to u u u.5. Substitute back to express the result in terms of x x x.

STEP 3

Perform the substitution u=7x+14 u = 7x + 14 u=7x+14.Calculate the derivative of u u u with respect to x x x:dudx=7 \frac{du}{dx} = 7 dxdu​=7Thus, dx=du7 dx = \frac{du}{7} dx=7du​.

STEP 4

Substitute u=7x+14 u = 7x + 14 u=7x+14 and dx=du7 dx = \frac{du}{7} dx=7du​ into the integral:∫7(7x+14)4 dx=∫7u4⋅du7 \int 7(7x+14)^4 \, dx = \int 7u^4 \cdot \frac{du}{7} ∫7(7x+14)4dx=∫7u4⋅7du​The 7 7 7 terms cancel out:∫u4 du \int u^4 \, du ∫u4du

STEP 5

Integrate with respect to u u u:∫u4 du=u55+C \int u^4 \, du = \frac{u^5}{5} + C ∫u4du=5u5​+Cwhere C C C is the constant of integration.

STEP 6

Substitute back u=7x+14 u = 7x + 14 u=7x+14 to express the result in terms of x x x:(7x+14)55+C \frac{(7x + 14)^5}{5} + C 5(7x+14)5​+CThus, the evaluated indefinite integral is:(7x+14)55+C \boxed{\frac{(7x + 14)^5}{5} + C} 5(7x+14)5​+C​

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QuestionMake the given substitution to evaluate the indefinite integral. $\begin{array}{l} \int 7(7 x+14)^{4} d x, u=7 x+14 \\ \int 7(7 x+14)^{4} d x=\square \end{array}$

1. We need to evaluate the indefinite integral using the given substitution.
2. The substitution $u = 7x + 14$ will simplify the integration process.
3. We will use the method of substitution to solve the integral.

1. Perform the substitution $u = 7x + 14$ .
2. Express $dx$ in terms of $du$ .
3. Substitute into the integral.
4. Integrate with respect to $u$ .
5. Substitute back to express the result in terms of $x$ .

Perform the substitution $u = 7x + 14$ .
Calculate the derivative of $u$ with respect to $x$ :
$\frac{du}{dx} = 7$
Thus, $dx = \frac{du}{7}$ .

Substitute $u = 7x + 14$ and $dx = \frac{du}{7}$ into the integral:
$\int 7(7x+14)^4 \, dx = \int 7u^4 \cdot \frac{du}{7}$
The $7$ terms cancel out:
$\int u^4 \, du$

Integrate with respect to $u$ :
$\int u^4 \, du = \frac{u^5}{5} + C$
where $C$ is the constant of integration.

Substitute back $u = 7x + 14$ to express the result in terms of $x$ :
$\frac{(7x + 14)^5}{5} + C$
Thus, the evaluated indefinite integral is:
$\boxed{\frac{(7x + 14)^5}{5} + C}$