What is the integral of sin(x)(cos(2x)) ?

The integral of sin(x)(cos(2x)) is 1/2 (-1/3 cos(3x)+cos(x))+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)(cos(2x))

\int sin (x) (cos (2 x)) d x

\frac{1}{2} (- \frac{1}{3} cos (3 x) + cos (x)) + C

Solution steps

\int sin (x) cos (2 x) d x

Use the following identity:

cos (t) sin (s) = \frac{sin ( s + t ) + sin ( s - t )}{2}

= \int \frac{sin ( x + 2 x ) + sin ( x - 2 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 sin (x + 2 x) + sin (x - 2 x) d x

Simplify

sin (x + 2 x) + sin (x - 2 x) : sin (3 x) - sin (x)

= \frac{1}{2} \cdot \int sin (3 x) - sin (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 sin (3 x) d x - \int sin (x) d x)

\int sin (3 x) d x = - \frac{1}{3} cos (3 x)

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

= \frac{1}{2} (- \frac{1}{3} cos (3 x) - (- cos (x)))

Simplify

= \frac{1}{2} (- \frac{1}{3} cos (3 x) + cos (x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{3} cos (3 x) + cos (x)) + C

n = 1 \sum \infty \frac{1}{1 + 4 ^{n}}

n = 0 \sum \infty a^{n}

\frac{d}{d x} ((5 x^{3} - 2 x^{2}) (6 x^{3}))

\frac{d}{d x} (1 - \frac{x ^{2}}{16})

y^{2} d x + x (x - y) d y = 0

What is the integral of sin(x)(cos(2x)) ?
The integral of sin(x)(cos(2x)) is 1/2 (-1/3 cos(3x)+cos(x))+C