Calculus Examples

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

Step 1

Let $u = t^{2}$ . Then $d u = 2 t d t$ , so $\frac{1}{2} d u = t d t$ . Rewrite using $u$ and $d$ $u$ .

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Step 1.1

Let $u = t^{2}$ . Find $\frac{d u}{d t}$ .

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Step 1.1.1

Differentiate $t^{2}$ .

$\frac{d}{d t} [t^{2}]$

Step 1.1.2

Differentiate using the Power Rule which states that $\frac{d}{d t} [t^{n}]$ is $n t^{n - 1}$ where $n = 2$ .

$2 t$

Step 1.2

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

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

Step 2

Combine $e^{u}$ and $\frac{1}{2}$ .

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

Step 3

Since $\frac{1}{2}$ is constant with respect to $u$ , move $\frac{1}{2}$ out of the integral.

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

Step 4

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

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

Step 5

Simplify.

$\frac{1}{2} e^{u} + C$

Step 6

Replace all occurrences of $u$ with $t^{2}$ .

$\frac{1}{2} e^{t^{2}} + C$