Calculus Examples

$\int x (e^{x^{2}} + 2) d x$

Step 1

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

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Step 1.1

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

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Step 1.1.1

Differentiate $x^{2}$ .

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

Step 1.1.2

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

$2 x$

Step 1.2

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

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

Step 2

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

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

Step 3

Split the single integral into multiple integrals.

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

Step 4

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

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

Step 5

Apply the constant rule.

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

Step 6

Simplify.

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

Step 7

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

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

Step 8

Simplify.

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Step 8.1

Apply the distributive property.

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

Step 8.2

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

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

Step 8.3

Cancel the common factor of $2$ .

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Step 8.3.1

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

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

Step 8.3.2

Cancel the common factor.

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

Step 8.3.3

Rewrite the expression.

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

Step 9

Reorder terms.

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