Calculus Examples

${(x^{4} + 9)}^{8} (4 x^{3} d x)$

Step 1

Simplify.

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Step 1.1

Raise $x$ to the power of $1$ .

$\int {(x^{4} + 9)}^{8} (4 d (x^{3} x^{1})) d x$

Step 1.2

Use the power rule $a^{m} a^{n} = a^{m + n}$ to combine exponents.

$\int {(x^{4} + 9)}^{8} (4 d x^{3 + 1}) d x$

Step 1.3

Add $3$ and $1$ .

$\int {(x^{4} + 9)}^{8} (4 d x^{4}) d x$

Step 1.4

Move $4$ to the left of ${(x^{4} + 9)}^{8}$ .

$\int 4 {(x^{4} + 9)}^{8} d x^{4} d x$

Step 2

Since $4 d$ is constant with respect to $x$ , move $4 d$ out of the integral.

$4 d \int {(x^{4} + 9)}^{8} x^{4} d x$

Step 3

Let $u_{1} = x^{4} + 9$ . Then $d u_{1} = 4 x^{3} d x$ , so $\frac{1}{4} d u_{1} = x^{3} d x$ . Rewrite using $u_{1}$ and $d$ $u_{1}$ .

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Step 3.1

Let $u_{1} = x^{4} + 9$ . Find $\frac{d u_{1}}{d x}$ .

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Step 3.1.1

Differentiate $x^{4} + 9$ .

$\frac{d}{d x} [x^{4} + 9]$

Step 3.1.2

By the Sum Rule, the derivative of $x^{4} + 9$ with respect to $x$ is $\frac{d}{d x} [x^{4}] + \frac{d}{d x} [9]$ .

$\frac{d}{d x} [x^{4}] + \frac{d}{d x} [9]$

Step 3.1.3

Differentiate using the Power Rule which states that $\frac{d}{d x} [x^{n}]$ is $n x^{n - 1}$ where $n = 4$ .

$4 x^{3} + \frac{d}{d x} [9]$

Step 3.1.4

Since $9$ is constant with respect to $x$ , the derivative of $9$ with respect to $x$ is $0$ .

$4 x^{3} + 0$

Step 3.1.5

Add $4 x^{3}$ and $0$ .

$4 x^{3}$

Step 3.2

Rewrite the problem using $u_{1}$ and $d u_{1}$ .

$4 d \int {u_{1}}^{8} \sqrt[4]{u_{1} - 9} \frac{1}{4} d u_{1}$

Step 4

Simplify.

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Step 4.1

Combine $\frac{1}{4}$ and ${u_{1}}^{8}$ .

$4 d \int \frac{{u_{1}}^{8}}{4} \sqrt[4]{u_{1} - 9} d u_{1}$

Step 4.2

Combine $\frac{{u_{1}}^{8}}{4}$ and $\sqrt[4]{u_{1} - 9}$ .

$4 d \int \frac{{u_{1}}^{8} \sqrt[4]{u_{1} - 9}}{4} d u_{1}$

Step 5

Since $\frac{1}{4}$ is constant with respect to $u_{1}$ , move $\frac{1}{4}$ out of the integral.

$4 d (\frac{1}{4} \int {u_{1}}^{8} \sqrt[4]{u_{1} - 9} d u_{1})$

Step 6

Simplify.

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Step 6.1

Combine $4$ and $\frac{1}{4}$ .

$d (\frac{4}{4} \int {u_{1}}^{8} \sqrt[4]{u_{1} - 9} d u_{1})$

Step 6.2

Combine $\frac{4}{4}$ and $d$ .

$\frac{4 d}{4} \int {u_{1}}^{8} \sqrt[4]{u_{1} - 9} d u_{1}$

Step 6.3

Cancel the common factor of $4$ .

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Step 6.3.1

Cancel the common factor.

$\frac{4 d}{4} \int {u_{1}}^{8} \sqrt[4]{u_{1} - 9} d u_{1}$

Step 6.3.2

Divide $d$ by $1$ .

$d \int {u_{1}}^{8} \sqrt[4]{u_{1} - 9} d u_{1}$

Step 7

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = {u_{1}}^{8}$ and $d v = \sqrt[4]{u_{1} - 9}$ .

$d ({u_{1}}^{8} (\frac{4}{5} {u_{2}}^{\frac{5}{4}}) - \int \frac{4}{5} {u_{2}}^{\frac{5}{4}} (8 {u_{1}}^{7}) d u_{1})$

Step 8

Simplify.

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Step 8.1

Combine $\frac{4}{5}$ and ${u_{2}}^{\frac{5}{4}}$ .

$d ({u_{1}}^{8} \frac{4 {u_{2}}^{\frac{5}{4}}}{5} - \int \frac{4}{5} {u_{2}}^{\frac{5}{4}} (8 {u_{1}}^{7}) d u_{1})$

Step 8.2

Combine ${u_{1}}^{8}$ and $\frac{4 {u_{2}}^{\frac{5}{4}}}{5}$ .

$d (\frac{{u_{1}}^{8} (4 {u_{2}}^{\frac{5}{4}})}{5} - \int \frac{4}{5} {u_{2}}^{\frac{5}{4}} (8 {u_{1}}^{7}) d u_{1})$

Step 8.3

Move $4$ to the left of ${u_{1}}^{8}$ .

$d (\frac{4 \cdot {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - \int \frac{4}{5} {u_{2}}^{\frac{5}{4}} (8 {u_{1}}^{7}) d u_{1})$

Step 8.4

Combine $\frac{4}{5}$ and ${u_{2}}^{\frac{5}{4}}$ .

$d (\frac{4 {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - \int \frac{4 {u_{2}}^{\frac{5}{4}}}{5} (8 {u_{1}}^{7}) d u_{1})$

Step 8.5

Combine $8$ and $\frac{4 {u_{2}}^{\frac{5}{4}}}{5}$ .

$d (\frac{4 {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - \int \frac{8 (4 {u_{2}}^{\frac{5}{4}})}{5} {u_{1}}^{7} d u_{1})$

Step 8.6

Multiply $4$ by $8$ .

$d (\frac{4 {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - \int \frac{32 {u_{2}}^{\frac{5}{4}}}{5} {u_{1}}^{7} d u_{1})$

Step 8.7

Combine $\frac{32 {u_{2}}^{\frac{5}{4}}}{5}$ and ${u_{1}}^{7}$ .

$d (\frac{4 {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - \int \frac{32 {u_{2}}^{\frac{5}{4}} {u_{1}}^{7}}{5} d u_{1})$

Step 9

Since $\frac{32 {u_{2}}^{\frac{5}{4}}}{5}$ is constant with respect to $u_{1}$ , move $\frac{32 {u_{2}}^{\frac{5}{4}}}{5}$ out of the integral.

$d (\frac{4 {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - (\frac{32 {u_{2}}^{\frac{5}{4}}}{5} \int {u_{1}}^{7} d u_{1}))$

Step 10

By the Power Rule, the integral of ${u_{1}}^{7}$ with respect to $u_{1}$ is $\frac{1}{8} {u_{1}}^{8}$ .

$d (\frac{4 {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - \frac{32 {u_{2}}^{\frac{5}{4}}}{5} (\frac{1}{8} {u_{1}}^{8} + C))$

Step 11

Simplify.

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Step 11.1

Combine $\frac{1}{8}$ and ${u_{1}}^{8}$ .

$d (\frac{4 {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - \frac{32 {u_{2}}^{\frac{5}{4}}}{5} (\frac{{u_{1}}^{8}}{8} + C))$

Step 11.2

Rewrite $d (\frac{4 {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - \frac{32 {u_{2}}^{\frac{5}{4}}}{5} (\frac{{u_{1}}^{8}}{8} + C))$ as $d (\frac{4 {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - \frac{4 {u_{2}}^{\frac{5}{4}} {u_{1}}^{8}}{5}) + C$ .

$d (\frac{4 {u_{1}}^{8} {u_{2}}^{\frac{5}{4}}}{5} - \frac{4 {u_{2}}^{\frac{5}{4}} {u_{1}}^{8}}{5}) + C$

Step 11.3

Simplify.

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Step 11.3.1

Reorder terms.

$d (\frac{4 {u_{2}}^{\frac{5}{4}} {u_{1}}^{8}}{5} - \frac{4 {u_{2}}^{\frac{5}{4}} {u_{1}}^{8}}{5}) + C$

Step 11.3.2

Subtract $\frac{4 {u_{2}}^{\frac{5}{4}} {u_{1}}^{8}}{5}$ from $\frac{4 {u_{2}}^{\frac{5}{4}} {u_{1}}^{8}}{5}$ .

$d \cdot 0 + C$

Step 11.3.3

Multiply $d$ by $0$ .

$0 + C$

Step 11.3.4

Add $0$ and $C$ .

$C$