What is the integral of sin^3(x)sin(3x) ?

The integral of sin^3(x)sin(3x) is

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integral of sin^3(x)sin(3x)

\int sin^{3} (x) sin (3 x) d x

- sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x) + C

Solution steps

\int sin^{3} (x) sin (3 x) d x

Apply Integration By Parts

: sin (3 x) (- cos (x) + \frac{1}{3} cos^{3} (x)) - \int - 3 cos (3 x) cos (x) + cos^{3} (x) cos (3 x) d x

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

\int - 3 cos (3 x) cos (x) + cos^{3} (x) cos (3 x) d x = \frac{1}{8} x - \frac{13}{32} sin (4 x) - \frac{23}{32} sin (2 x) + \frac{1}{8} cos^{2} (x) sin (4 x) + \frac{1}{4} cos^{2} (x) sin (2 x) - \frac{1}{96} sin (6 x)

= sin (3 x) (- cos (x) + \frac{1}{3} cos^{3} (x)) - (\frac{1}{8} x - \frac{13}{32} sin (4 x) - \frac{23}{32} sin (2 x) + \frac{1}{8} cos^{2} (x) sin (4 x) + \frac{1}{4} cos^{2} (x) sin (2 x) - \frac{1}{96} sin (6 x))

Simplify

sin (3 x) (- cos (x) + \frac{1}{3} cos^{3} (x)) - (\frac{1}{8} x - \frac{13}{32} sin (4 x) - \frac{23}{32} sin (2 x) + \frac{1}{8} cos^{2} (x) sin (4 x) + \frac{1}{4} cos^{2} (x) sin (2 x) - \frac{1}{96} sin (6 x)) : - sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x)

= - sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x)

Add a constant to the solution

= - sin (3 x) cos (x) + \frac{1}{3} cos^{3} (x) sin (3 x) - \frac{1}{8} x + \frac{13}{32} sin (4 x) + \frac{23}{32} sin (2 x) - \frac{1}{8} cos^{2} (x) sin (4 x) - \frac{1}{4} cos^{2} (x) sin (2 x) + \frac{1}{96} sin (6 x) + C

\int \frac{3 sin ( x )}{( 1 + cos ( x ) ) ^{2}} d x

\frac{d}{d x} ((ln (x)) (arctan (x)))

n = 1 \sum \infty \frac{2 ^{n}}{n 3 ^{n}}

\int \frac{24}{1 + 16 x ^{2}} d x

n = 0 \sum \infty (\frac{1}{2})^{n} x^{n}