How do you integrate $\int x cos (3 x)$ $int xcos(3x)$ by integration by parts method?

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How do you integrate $\int x cos (3 x)$ $int xcos(3x)$ by integration by parts method?

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Answer 1 · 2016-12-31T02:20:02

The goal of integration by parts is to differentiate one of the terms in the product so that it becomes $1$ $1$ , leaving only one term to be integrated. Take a look at this problem to understand better.

Let $u = x$ $u = x$ and $d v = cos (3 x) d x$ $dv = cos(3x)dx$ . To apply the integration by parts formula, we need $v$ $v$ and $d u$ $du$ . $d u = d x$ $du = dx$ . We can find $v$ $v$ through $u$ $u$ -substitution.

Let $u_{2} = 3 x$ $u_2 = 3x$ . Then $d u_{2} = 3 d x$ $du_2 = 3dx$ and $d x = \frac{d u_{2}}{3}$ $dx = (du_2)/3$

We can rewrite:

$= \int ({cos u}_{2}) \cdot \frac{d u_{2}}{3}$ $=int(cosu_2) * (du_2)/3$

$= \frac{1}{3} \int {cos u}_{2} d u_{2}$ $=1/3intcosu_2du_2$

$= \frac{1}{3} {sin u}_{2}$ $=1/3sinu_2$

Since $u_{2} = 3 x$ $u_2 = 3x$ :

$= \frac{1}{3} sin (3 x)$ $=1/3sin(3x)$

Thus, $v = \frac{1}{3} sin (3 x)$ $v = 1/3sin(3x)$ .

The integration by parts formula is $\int (u d v) = u v - \int (v d u)$ $int(udv) = uv - int(vdu)$ .

$\int (x cos (3 x)) d x = \frac{1}{3} sin (3 x) \cdot x - \int (1 \cdot \frac{1}{3} sin (3 x) d x)$ $int(xcos(3x)) dx= 1/3sin(3x) * x - int(1 * 1/3sin(3x)dx)$

$\int (x cos 3 x) d x = \frac{1}{3} x sin (3 x) - \int (\frac{1}{3} sin (3 x) d x)$ $int(xcos3x)dx = 1/3xsin(3x) - int(1/3sin(3x)dx)$

We repeat the substitution process performed above to integrate $\frac{1}{3} sin (3 x)$ $1/3sin(3x)$

Let $u = 3 x$ $u = 3x$ . Then $d u = 3 d x$ $du = 3dx$ and $d x = \frac{1}{3} d u$ $dx = 1/3du$

$= \int \frac{1}{3} sin u \cdot \frac{1}{3} d u$ $=int1/3sinu * 1/3du$

$= \frac{1}{9} \int sin u d u$ $=1/9intsinu du$

$= - \frac{1}{9} cos u$ $=-1/9cosu$

$= - \frac{1}{9} cos (3 x)$ $=-1/9cos(3x)$

The integral of the entire expression is therefore:

$\int (x cos 3 x) d x = \frac{1}{3} x sin 3 x + \frac{1}{9} cos 3 x + C$ $int(xcos3x)dx = 1/3xsin3x+ 1/9cos3x+ C$ , where $C$ $C$ is a constant.

Hopefully this helps!

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