Calculus Examples

$\int x^{6} d x$

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 $d u$ .

$\int {\sqrt{u}}^{5} \frac{1}{2} d u$

Step 2

Simplify.

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Step 2.1

Rewrite ${\sqrt{u}}^{5}$ as $\sqrt{u^{5}}$ .

$\int \sqrt{u^{5}} \frac{1}{2} d u$

Step 2.2

Combine $\sqrt{u^{5}}$ and $\frac{1}{2}$ .

$\int \frac{\sqrt{u^{5}}}{2} d u$

Step 3

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

$\frac{1}{2} \int \sqrt{u^{5}} d u$

Step 4

Use $\sqrt[n]{a^{x}} = a^{\frac{x}{n}}$ to rewrite $\sqrt{u^{5}}$ as $u^{\frac{5}{2}}$ .

$\frac{1}{2} \int u^{\frac{5}{2}} d u$

Step 5

By the Power Rule, the integral of $u^{\frac{5}{2}}$ with respect to $u$ is $\frac{2}{7} u^{\frac{7}{2}}$ .

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

Step 6

Simplify.

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Step 6.1

Rewrite $\frac{1}{2} (\frac{2}{7} u^{\frac{7}{2}} + C)$ as $\frac{1}{2} \cdot \frac{2}{7} u^{\frac{7}{2}} + C$ .

$\frac{1}{2} \cdot \frac{2}{7} u^{\frac{7}{2}} + C$

Step 6.2

Rewrite $\frac{1}{2} \cdot \frac{2}{7} u^{\frac{7}{2}} + C$ as $\frac{1}{7} u^{\frac{7}{2}} + C$ .

$\frac{1}{7} u^{\frac{7}{2}} + C$

Step 7

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

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