What is the integral of sin(4x)sin(x) ?

The integral of sin(4x)sin(x) is 1/2 (-1/5 sin(5x)+1/3 sin(3x))+C

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integral of sin(4x)sin(x)

\int sin (4 x) sin (x) d x

\frac{1}{2} (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) + C

Solution steps

\int sin (4 x) sin (x) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (5 x) + cos (3 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 (5 x) + cos (3 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 (5 x) d x + \int cos (3 x) d x)

\int cos (5 x) d x = \frac{1}{5} sin (5 x)

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

= \frac{1}{2} (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x))

Add a constant to the solution

= \frac{1}{2} (- \frac{1}{5} sin (5 x) + \frac{1}{3} sin (3 x)) + C

2 (t + 2)

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

arccos (3 x)

\int u sin (u) d u

x y + y^{^{'}} = 2 x, y (0) = 5

What is the integral of sin(4x)sin(x) ?
The integral of sin(4x)sin(x) is 1/2 (-1/5 sin(5x)+1/3 sin(3x))+C