Calculus Examples

$x \cos (5 x)$

Step 1

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

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

Step 2

Simplify.

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

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

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

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

Step 3

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

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

Step 4

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

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Differentiate $5 x$ .

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

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

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

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

$5 \cdot 1$

Multiply $5$ by $1$ .

$5$

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

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

Step 5

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

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

Step 6

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

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

Step 7

Simplify.

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

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

Multiply $5$ by $5$ .

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

Step 8

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

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

Step 9

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

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

Step 10

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

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

Step 11

Reorder terms.

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