What is the integral of sin(15x)sin(16x) ?

The integral of sin(15x)sin(16x) is 1/2 (sin(x)-1/31 sin(31x))+C

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

integral of sin(15x)sin(16x)

\int sin (15 x) sin (16 x) d x

\frac{1}{2} (sin (x) - \frac{1}{31} sin (31 x)) + C

Solution steps

\int sin (15 x) sin (16 x) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (31 x) + cos (- 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 - cos (31 x) + cos (- x) d x

Simplify

- cos (31 x) + cos (- x) : cos (x) - cos (31 x)

= \frac{1}{2} \cdot \int cos (x) - cos (31 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) d x - \int cos (31 x) d x)

\int cos (x) d x = sin (x)

\int cos (31 x) d x = \frac{1}{31} sin (31 x)

= \frac{1}{2} (sin (x) - \frac{1}{31} sin (31 x))

Add a constant to the solution

= \frac{1}{2} (sin (x) - \frac{1}{31} sin (31 x)) + C

\int \frac{a}{( x ^{2} + a ^{2} ) ^{\frac{3}{2}}} d x

\int (x^{3} - 2)^{2} d x

\int \frac{12 x ^{2} - 6 x + 2}{( 4 x + 1 ) ( x ^{2} + 1 )} d x

\int \frac{1}{( 3 x + 1 ) ^{6}} d x

\int cos (13 - 2 y) d y

What is the integral of sin(15x)sin(16x) ?
The integral of sin(15x)sin(16x) is 1/2 (sin(x)-1/31 sin(31x))+C