Calculus Examples

$\int t^{3} + e^{5 t} d t$

Step 1

Split the single integral into multiple integrals.

$\int t^{3} d t + \int e^{5 t} d t$

Step 2

By the Power Rule, the integral of $t^{3}$ with respect to $t$ is $\frac{1}{4} t^{4}$ .

$\frac{1}{4} t^{4} + C + \int e^{5 t} d t$

Step 3

Let $u = 5 t$ . Then $d u = 5 d t$ , so $\frac{1}{5} d u = d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 3.1

Let $u = 5 t$ . Find $\frac{d u}{d t}$ .

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Step 3.1.1

Differentiate $5 t$ .

$\frac{d}{d t} [5 t]$

Step 3.1.2

Since $5$ is constant with respect to $t$ , the derivative of $5 t$ with respect to $t$ is $5 \frac{d}{d t} [t]$ .

$5 \frac{d}{d t} [t]$

Step 3.1.3

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

$5 \cdot 1$

Step 3.1.4

Multiply $5$ by $1$ .

$5$

Step 3.2

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

$\frac{1}{4} t^{4} + C + \int e^{u} \frac{1}{5} d u$

Step 4

Combine $e^{u}$ and $\frac{1}{5}$ .

$\frac{1}{4} t^{4} + C + \int \frac{e^{u}}{5} d u$

Step 5

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$\frac{1}{4} t^{4} + C + \frac{1}{5} \int e^{u} d u$

Step 6

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

$\frac{1}{4} t^{4} + C + \frac{1}{5} (e^{u} + C)$

Step 7

Simplify.

$\frac{1}{4} t^{4} + \frac{1}{5} e^{u} + C$

Step 8

Replace all occurrences of $u$ with $5 t$ .

$\frac{1}{4} t^{4} + \frac{1}{5} e^{5 t} + C$