Calculus Examples

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

Step 1

Split the single integral into multiple integrals.

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

Step 2

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

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Step 2.1

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

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Step 2.1.1

Differentiate $2 x$ .

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

Step 2.1.2

Since $2$ is constant with respect to $x$ , the derivative of $2 x$ with respect to $x$ is $2 \frac{d}{d x} [x]$ .

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

Step 2.1.3

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

$2 \cdot 1$

Step 2.1.4

Multiply $2$ by $1$ .

$2$

Step 2.2

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

$\int \sin (u) \frac{1}{2} d u + \int \cos (x) d x$

Step 3

Combine $\sin (u)$ and $\frac{1}{2}$ .

$\int \frac{\sin (u)}{2} d u + \int \cos (x) d x$

Step 4

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

$\frac{1}{2} \int \sin (u) d u + \int \cos (x) d x$

Step 5

The integral of $\sin (u)$ with respect to $u$ is $- \cos (u)$ .

$\frac{1}{2} (- \cos (u) + C) + \int \cos (x) d x$

Step 6

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

$\frac{1}{2} (- \cos (u) + C) + \sin (x) + C$

Step 7

Simplify.

$- \frac{\cos (u)}{2} + \sin (x) + C$

Step 8

Replace all occurrences of $u$ with $2 x$ .

$- \frac{\cos (2 x)}{2} + \sin (x) + C$

Step 9

Reorder terms.

$- \frac{1}{2} \cos (2 x) + \sin (x) + C$