Calculus Examples

$\int 8 e^{4 y} d y$

Step 1

Since $8$ is constant with respect to $y$ , move $8$ out of the integral.

$8 \int e^{4 y} d y$

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $4 y$ .

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

Step 2.1.2

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

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

Step 2.1.3

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

$4 \cdot 1$

Step 2.1.4

Multiply $4$ by $1$ .

$4$

Step 2.2

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

$8 \int e^{u} \frac{1}{4} d u$

Step 3

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

$8 \int \frac{e^{u}}{4} d u$

Step 4

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

$8 (\frac{1}{4} \int e^{u} d u)$

Step 5

Simplify.

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Step 5.1

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

$\frac{8}{4} \int e^{u} d u$

Step 5.2

Cancel the common factor of $8$ and $4$ .

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Step 5.2.1

Factor $4$ out of $8$ .

$\frac{4 \cdot 2}{4} \int e^{u} d u$

Step 5.2.2

Cancel the common factors.

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Step 5.2.2.1

Factor $4$ out of $4$ .

$\frac{4 \cdot 2}{4 (1)} \int e^{u} d u$

Step 5.2.2.2

Cancel the common factor.

$\frac{4 \cdot 2}{4 \cdot 1} \int e^{u} d u$

Step 5.2.2.3

Rewrite the expression.

$\frac{2}{1} \int e^{u} d u$

Step 5.2.2.4

Divide $2$ by $1$ .

$2 \int e^{u} d u$

Step 6

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

$2 (e^{u} + C)$

Step 7

Simplify.

$2 e^{u} + C$

Step 8

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

$2 e^{4 y} + C$