Integrate x 2 − 5 x + cos ⁡ ( x ) 2 − 3 cos ⁡ ( 3 x ) with respect to x.

Question

Integrate       x    2        ﻿    −      5    x    +            cos      ⁡      (      x      )        2    −  3  cos  ⁡  (  3  x  ) with respect to x.

Bumanro5mv · Accepted Answer

Split the single integral into multiple integrals.

$\int \frac{x}{2} d x + \int - \frac{5}{x} d x + \int \frac{\cos (x)}{2} d x + \int - 3 \cos (3 x) d x$

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

$\frac{1}{2} \int x d x + \int - \frac{5}{x} d x + \int \frac{\cos (x)}{2} d x + \int - 3 \cos (3 x) d x$

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

$\frac{1}{2} (\frac{1}{2} x^{2} + C) + \int - \frac{5}{x} d x + \int \frac{\cos (x)}{2} d x + \int - 3 \cos (3 x) d x$

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

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - \int \frac{5}{x} d x + \int \frac{\cos (x)}{2} d x + \int - 3 \cos (3 x) d x$

Since $5$ is constant with respect to $x$ , move $5$ out of the integral.

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - (5 \int \frac{1}{x} d x) + \int \frac{\cos (x)}{2} d x + \int - 3 \cos (3 x) d x$

Multiply $5$ by $- 1$ .

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 5 \int \frac{1}{x} d x + \int \frac{\cos (x)}{2} d x + \int - 3 \cos (3 x) d x$

The integral of $\frac{1}{x}$ with respect to $x$ is $\ln (| x |)$ .

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 5 (\ln (| x |) + C) + \int \frac{\cos (x)}{2} d x + \int - 3 \cos (3 x) d x$

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

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 5 (\ln (| x |) + C) + \frac{1}{2} \int \cos (x) d x + \int - 3 \cos (3 x) d x$

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

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 5 (\ln (| x |) + C) + \frac{1}{2} (\sin (x) + C) + \int - 3 \cos (3 x) d x$

Since $- 3$ is constant with respect to $x$ , move $- 3$ out of the integral.

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 5 (\ln (| x |) + C) + \frac{1}{2} (\sin (x) + C) - 3 \int \cos (3 x) d x$

Let $u = 3 x$ . Then $d u = 3 d x$ , so $\frac{1}{3} d u = d x$ . Rewrite using $u$ and $d$ $u$ .

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 5 (\ln (| x |) + C) + \frac{1}{2} (\sin (x) + C) - 3 \int \cos (u) \frac{1}{3} d u$

Combine $\cos (u)$ and $\frac{1}{3}$ .

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 5 (\ln (| x |) + C) + \frac{1}{2} (\sin (x) + C) - 3 \int \frac{\cos (u)}{3} d u$

Since $\frac{1}{3}$ is constant with respect to $u$ , move $\frac{1}{3}$ out of the integral.

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 5 (\ln (| x |) + C) + \frac{1}{2} (\sin (x) + C) - 3 (\frac{1}{3} \int \cos (u) d u)$

Simplify.

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 5 (\ln (| x |) + C) + \frac{1}{2} (\sin (x) + C) - \int \cos (u) d u$

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

$\frac{1}{2} (\frac{1}{2} x^{2} + C) - 5 (\ln (| x |) + C) + \frac{1}{2} (\sin (x) + C) - (\sin (u) + C)$

Simplify.

$\frac{x^{2}}{4} - 5 \ln (| x |) + \frac{\sin (x)}{2} - \sin (u) + C$

Replace all occurrences of $u$ with $3 x$ .

$\frac{x^{2}}{4} - 5 \ln (| x |) + \frac{\sin (x)}{2} - \sin (3 x) + C$

Reorder terms.

$\frac{1}{4} x^{2} - 5 \ln (| x |) + \frac{1}{2} \sin (x) - \sin (3 x) + C$

Integrate x 2  </mrow> − 5 x + <mrow>

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