What is the integral of xsin^2(5x) ?

The integral of xsin^2(5x) is 1/200 (50x^2-10xsin(10x)-cos(10x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of xsin^2(5x)

\int x sin^{2} (5 x) d x

\frac{1}{200} (50 x^{2} - 10 x sin (10 x) - cos (10 x)) + C

Solution steps

\int x sin^{2} (5 x) d x

Rewrite using trig identities

= \int \frac{1}{2} x (1 - cos (10 x)) d x

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{1}{2} \cdot \int x (1 - cos (10 x)) d x

Apply Integration By Parts

: x (x - \frac{1}{10} sin (10 x)) - \int x - \frac{1}{10} sin (10 x) d x

= \frac{1}{2} (x (x - \frac{1}{10} sin (10 x)) - \int x - \frac{1}{10} sin (10 x) d x)

\int x - \frac{1}{10} sin (10 x) d x = \frac{x ^{2}}{2} + \frac{1}{100} cos (10 x)

= \frac{1}{2} (x (x - \frac{1}{10} sin (10 x)) - (\frac{x ^{2}}{2} + \frac{1}{100} cos (10 x)))

Simplify

\frac{1}{2} (x (x - \frac{1}{10} sin (10 x)) - (\frac{x ^{2}}{2} + \frac{1}{100} cos (10 x))) : \frac{1}{200} (50 x^{2} - 10 x sin (10 x) - cos (10 x))

= \frac{1}{200} (50 x^{2} - 10 x sin (10 x) - cos (10 x))

Add a constant to the solution

= \frac{1}{200} (50 x^{2} - 10 x sin (10 x) - cos (10 x)) + C

\frac{d y}{d t} = 1 y

\int \frac{x + 2}{x + 1} d x

11 cos (- x)

\frac{d}{d x} (3 sin (x) - sin (3 x))

x \to + 0 lim (\frac{- 1}{5 x})

What is the integral of xsin^2(5x) ?
The integral of xsin^2(5x) is 1/200 (50x^2-10xsin(10x)-cos(10x))+C