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

The integral of sin(nx)cos(nx) is -1/(2n)cos^2(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(nx)cos(nx)

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

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

Solution steps

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

Apply u-substitution

: \int - \frac{u}{n} d u

= \int - \frac{u}{n} d u

Take the constant out:

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

= - \frac{1}{n} \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

= - \frac{1}{n} \cdot \frac{u ^{2}}{2}

Substitute back

u = cos (n x)

= - \frac{1}{n} \cdot \frac{cos ^{2} ( n x )}{2}

Simplify

- \frac{1}{n} \cdot \frac{cos ^{2} ( n x )}{2} : - \frac{1}{2 n} cos^{2} (n x)

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

Add a constant to the solution

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

\int - 2 x arctan (x) d x

\int \frac{x ^{6}}{( 9 - x ^{7} ) ^{2}} d x

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

\int \frac{1}{x ln ( x ) ln ( x )} d x

\int 2 x (x^{2} - 1)^{99} d x

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