What is the integral of sin(x)sin(nx) ?

The integral of sin(x)sin(nx) is 1/2 (-1/(1+n)sin(x+nx)+1/(1-n)sin(x-nx))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin(x)sin(nx)

\int sin (x) sin (n x) d x

\frac{1}{2} (- \frac{1}{1 + n} sin (x + n x) + \frac{1}{1 - n} sin (x - n x)) + C

Solution steps

\int sin (x) sin (n x) d x

Rewrite using trig identities

= \int \frac{- cos ( x + n x ) + cos ( x - n x )}{2} 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 - cos (x + n x) + cos (x - n x) d x

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int cos (x + n x) d x = \frac{1}{1 + n} sin (x + n x)

\int cos (x - n x) d x = \frac{1}{1 - n} sin (x - n x)

= \frac{1}{2} (- \frac{1}{1 + n} sin (x + n x) + \frac{1}{1 - n} sin (x - n x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{1 + n} sin (x + n x) + \frac{1}{1 - n} sin (x - n x)) + C

sin (x + y) = 2 x - 2 y, (π, π)

\frac{d}{d x} (lo g_{10} (x^{3}))

y = x^{3} - 15 x, (2, - 22)

9 + 5 x^{2} - 2 x^{3}

\int \frac{x ^{2} - 1}{x ^{4}} d x

What is the integral of sin(x)sin(nx) ?
The integral of sin(x)sin(nx) is 1/2 (-1/(1+n)sin(x+nx)+1/(1-n)sin(x-nx))+C