Calculus Examples

$\int (7 x + e^{4 x}) d x$

Step 1

Let $u = 4 x$ . Then $d u = 4 d x$ , so $\frac{1}{4} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = 4 x$ . Find $\frac{d u}{d x}$ .

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Step 1.1.1

Differentiate $4 x$ .

$\frac{d}{d x} [4 x]$

Step 1.1.2

Since $4$ is constant with respect to $x$ , the derivative of $4 x$ with respect to $x$ is $4 \frac{d}{d x} [x]$ .

$4 \frac{d}{d x} [x]$

Step 1.1.3

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

$4 \cdot 1$

Step 1.1.4

Multiply $4$ by $1$ .

$4$

Step 1.2

Rewrite the problem using $u$ and $d u$ .

$\int (7 \frac{u}{4} + e^{u}) \frac{1}{4} d u$

Step 2

Combine $7$ and $\frac{u}{4}$ .

$\int (\frac{7 u}{4} + e^{u}) \frac{1}{4} d u$

Step 3

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

$\frac{1}{4} \int \frac{7 u}{4} + e^{u} d u$

Step 4

Split the single integral into multiple integrals.

$\frac{1}{4} (\int \frac{7 u}{4} d u + \int e^{u} d u)$

Step 5

Since $\frac{7}{4}$ is constant with respect to $u$ , move $\frac{7}{4}$ out of the integral.

$\frac{1}{4} (\frac{7}{4} \int u d u + \int e^{u} d u)$

Step 6

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

$\frac{1}{4} (\frac{7}{4} (\frac{1}{2} u^{2} + C) + \int e^{u} d u)$

Step 7

The integral of $e^{u}$ with respect to $u$ is $e^{u}$ .

$\frac{1}{4} (\frac{7}{4} (\frac{1}{2} u^{2} + C) + e^{u} + C)$

Step 8

Simplify.

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Step 8.1

Simplify.

$\frac{1}{4} (\frac{7 u^{2}}{8} + e^{u}) + C$

Step 8.2

Reorder terms.

$\frac{1}{4} (\frac{7}{8} u^{2} + e^{u}) + C$

Step 9

Replace all occurrences of $u$ with $4 x$ .

$\frac{1}{4} (\frac{7}{8} {(4 x)}^{2} + e^{4 x}) + C$

Step 10

Simplify.

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Step 10.1

Simplify each term.

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Step 10.1.1

Apply the product rule to $4 x$ .

$\frac{1}{4} (\frac{7}{8} (4^{2} x^{2}) + e^{4 x}) + C$

Step 10.1.2

Raise $4$ to the power of $2$ .

$\frac{1}{4} (\frac{7}{8} (16 x^{2}) + e^{4 x}) + C$

Step 10.1.3

Cancel the common factor of $8$ .

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Step 10.1.3.1

Factor $8$ out of $16 x^{2}$ .

$\frac{1}{4} (\frac{7}{8} (8 (2 x^{2})) + e^{4 x}) + C$

Step 10.1.3.2

Cancel the common factor.

$\frac{1}{4} (\frac{7}{8} (8 (2 x^{2})) + e^{4 x}) + C$

Step 10.1.3.3

Rewrite the expression.

$\frac{1}{4} (7 (2 x^{2}) + e^{4 x}) + C$

Step 10.1.4

Multiply $2$ by $7$ .

$\frac{1}{4} (14 x^{2} + e^{4 x}) + C$

Step 10.2

Apply the distributive property.

$\frac{1}{4} (14 x^{2}) + \frac{1}{4} e^{4 x} + C$

Step 10.3

Cancel the common factor of $2$ .

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Step 10.3.1

Factor $2$ out of $4$ .

$\frac{1}{2 (2)} (14 x^{2}) + \frac{1}{4} e^{4 x} + C$

Step 10.3.2

Factor $2$ out of $14 x^{2}$ .

$\frac{1}{2 (2)} (2 (7 x^{2})) + \frac{1}{4} e^{4 x} + C$

Step 10.3.3

Cancel the common factor.

$\frac{1}{2 \cdot 2} (2 (7 x^{2})) + \frac{1}{4} e^{4 x} + C$

Step 10.3.4

Rewrite the expression.

$\frac{1}{2} (7 x^{2}) + \frac{1}{4} e^{4 x} + C$

Step 10.4

Combine $7$ and $\frac{1}{2}$ .

$\frac{7}{2} x^{2} + \frac{1}{4} e^{4 x} + C$

Step 10.5

Combine $\frac{7}{2}$ and $x^{2}$ .

$\frac{7 x^{2}}{2} + \frac{1}{4} e^{4 x} + C$

Step 10.6

Combine $\frac{1}{4}$ and $e^{4 x}$ .

$\frac{7 x^{2}}{2} + \frac{e^{4 x}}{4} + C$

Step 11

Reorder terms.

$\frac{7}{2} x^{2} + \frac{1}{4} e^{4 x} + C$