Calculus Examples

$- (x + 1) \sin (5 x)$

Step 1

Since $- 1$ is constant with respect to $x$ , move $- 1$ out of the integral.

$- \int (x + 1) \sin (5 x) d x$

Step 2

Integrate by parts using the formula $\int u d v = u v - \int v d u$ , where $u = x + 1$ and $d v = \sin (5 x)$ .

$- ((x + 1) (- \frac{1}{5} \cos (5 x)) - \int - \frac{1}{5} \cos (5 x) d x)$

Step 3

Since $- \frac{1}{5}$ is constant with respect to $x$ , move $- \frac{1}{5}$ out of the integral.

$- ((x + 1) (- \frac{1}{5} \cos (5 x)) - (- \frac{1}{5} \int \cos (5 x) d x))$

Step 4

Simplify.

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Step 4.1

Combine $\cos (5 x)$ and $\frac{1}{5}$ .

$- ((x + 1) (- \frac{\cos (5 x)}{5}) - (- \frac{1}{5} \int \cos (5 x) d x))$

Step 4.2

Multiply $- 1$ by $- 1$ .

$- ((x + 1) (- \frac{\cos (5 x)}{5}) + 1 (\frac{1}{5} \int \cos (5 x) d x))$

Step 4.3

Multiply $\frac{1}{5}$ by $1$ .

$- ((x + 1) (- \frac{\cos (5 x)}{5}) + \frac{1}{5} \int \cos (5 x) d x)$

Step 5

Let $u = 5 x$ . Then $d u = 5 d x$ , so $\frac{1}{5} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

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Step 5.1

Let $u = 5 x$ . Find $\frac{d u}{d x}$ .

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Step 5.1.1

Differentiate $5 x$ .

$\frac{d}{d x} [5 x]$

Step 5.1.2

Since $5$ is constant with respect to $x$ , the derivative of $5 x$ with respect to $x$ is $5 \frac{d}{d x} [x]$ .

$5 \frac{d}{d x} [x]$

Step 5.1.3

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

$5 \cdot 1$

Step 5.1.4

Multiply $5$ by $1$ .

$5$

Step 5.2

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

$- ((x + 1) (- \frac{\cos (5 x)}{5}) + \frac{1}{5} \int \cos (u) \frac{1}{5} d u)$

Step 6

Combine $\cos (u)$ and $\frac{1}{5}$ .

$- ((x + 1) (- \frac{\cos (5 x)}{5}) + \frac{1}{5} \int \frac{\cos (u)}{5} d u)$

Step 7

Since $\frac{1}{5}$ is constant with respect to $u$ , move $\frac{1}{5}$ out of the integral.

$- ((x + 1) (- \frac{\cos (5 x)}{5}) + \frac{1}{5} (\frac{1}{5} \int \cos (u) d u))$

Step 8

Simplify.

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Step 8.1

Multiply $\frac{1}{5}$ by $\frac{1}{5}$ .

$- ((x + 1) (- \frac{\cos (5 x)}{5}) + \frac{1}{5 \cdot 5} \int \cos (u) d u)$

Step 8.2

Multiply $5$ by $5$ .

$- ((x + 1) (- \frac{\cos (5 x)}{5}) + \frac{1}{25} \int \cos (u) d u)$

Step 9

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

$- ((x + 1) (- \frac{\cos (5 x)}{5}) + \frac{1}{25} (\sin (u) + C))$

Step 10

Rewrite $- ((x + 1) (- \frac{\cos (5 x)}{5}) + \frac{1}{25} (\sin (u) + C))$ as $- (- \frac{\cos (5 x) x}{5} - \frac{\cos (5 x)}{5} + \frac{\sin (u)}{25}) + C$ .

$- (- \frac{\cos (5 x) x}{5} - \frac{\cos (5 x)}{5} + \frac{\sin (u)}{25}) + C$

Step 11

Replace all occurrences of $u$ with $5 x$ .

$- (- \frac{\cos (5 x) x}{5} - \frac{\cos (5 x)}{5} + \frac{\sin (5 x)}{25}) + C$

Step 12

Simplify.

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Step 12.1

Apply the distributive property.

$- - \frac{\cos (5 x) x}{5} - - \frac{\cos (5 x)}{5} - \frac{\sin (5 x)}{25} + C$

Step 12.2

Simplify.

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Step 12.2.1

Multiply $- - \frac{\cos (5 x) x}{5}$ .

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Step 12.2.1.1

Multiply $- 1$ by $- 1$ .

$1 \frac{\cos (5 x) x}{5} - - \frac{\cos (5 x)}{5} - \frac{\sin (5 x)}{25} + C$

Step 12.2.1.2

Multiply $\frac{\cos (5 x) x}{5}$ by $1$ .

$\frac{\cos (5 x) x}{5} - - \frac{\cos (5 x)}{5} - \frac{\sin (5 x)}{25} + C$

Step 12.2.2

Multiply $- - \frac{\cos (5 x)}{5}$ .

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Step 12.2.2.1

Multiply $- 1$ by $- 1$ .

$\frac{\cos (5 x) x}{5} + 1 \frac{\cos (5 x)}{5} - \frac{\sin (5 x)}{25} + C$

Step 12.2.2.2

Multiply $\frac{\cos (5 x)}{5}$ by $1$ .

$\frac{\cos (5 x) x}{5} + \frac{\cos (5 x)}{5} - \frac{\sin (5 x)}{25} + C$

Step 12.3

Reorder factors in $\frac{\cos (5 x) x}{5} + \frac{\cos (5 x)}{5} - \frac{\sin (5 x)}{25}$ .

$\frac{x \cos (5 x)}{5} + \frac{\cos (5 x)}{5} - \frac{\sin (5 x)}{25} + C$

Step 13

Reorder terms.

$\frac{1}{5} x \cos (5 x) + \frac{1}{5} \cos (5 x) - \frac{1}{25} \sin (5 x) + C$