Calculus Examples

$\int 2 \sin (x) \cos (x) d x$

Step 1

Since $2$ is constant with respect to $x$ , move $2$ out of the integral.

$2 \int \sin (x) \cos (x) d x$

Step 2

Let $u = \sin (x)$ . Then $d u = \cos (x) d x$ , so $\frac{1}{\cos (x)} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 2.1

Let $u = \sin (x)$ . Find $\frac{d u}{d x}$ .

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Step 2.1.1

Differentiate $\sin (x)$ .

$\frac{d}{d x} [\sin (x)]$

Step 2.1.2

The derivative of $\sin (x)$ with respect to $x$ is $\cos (x)$ .

$\cos (x)$

Step 2.2

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

$2 \int u d u$

Step 3

By the Power Rule, the integral of $u$ with respect to $u$ is $\frac{1}{2} u^{2}$ .

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

Step 4

Simplify.

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Step 4.1

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

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

Step 4.2

Simplify.

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Step 4.2.1

Combine $2$ and $\frac{1}{2}$ .

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

Step 4.2.2

Cancel the common factor of $2$ .

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Step 4.2.2.1

Cancel the common factor.

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

Step 4.2.2.2

Rewrite the expression.

$1 u^{2} + C$

Step 4.2.3

Multiply $u^{2}$ by $1$ .

$u^{2} + C$

Step 5

Replace all occurrences of $u$ with $\sin (x)$ .

$\sin^{2} (x) + C$