What is the integral of sin(1/5 x)sin(1/10 x) ?

The integral of sin(1/5 x)sin(1/10 x) is 1/2 (-10/3 sin((3x)/(10))+10sin(x/(10)))+C

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integral of sin(1/5 x)sin(1/10 x)

\int sin (\frac{1}{5} x) sin (\frac{1}{10} x) d x

\frac{1}{2} (- \frac{10}{3} sin (\frac{3 x}{10}) + 10 sin (\frac{x}{10})) + C

Solution steps

\int sin (\frac{1}{5} x) sin (\frac{1}{10} x) d x

Simplify

sin (\frac{1}{5} x) sin (\frac{1}{10} x) : sin (\frac{x}{5}) sin (\frac{x}{10})

= \int sin (\frac{x}{5}) sin (\frac{x}{10}) d x

Rewrite using trig identities

= \int \frac{1}{2} (- cos (\frac{3 x}{10}) + cos (\frac{x}{10})) 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 (\frac{3 x}{10}) + cos (\frac{x}{10}) 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 (\frac{3 x}{10}) d x + \int cos (\frac{x}{10}) d x)

\int cos (\frac{3 x}{10}) d x = \frac{10}{3} sin (\frac{3 x}{10})

\int cos (\frac{x}{10}) d x = 10 sin (\frac{x}{10})

= \frac{1}{2} (- \frac{10}{3} sin (\frac{3 x}{10}) + 10 sin (\frac{x}{10}))

Add a constant to the solution

= \frac{1}{2} (- \frac{10}{3} sin (\frac{3 x}{10}) + 10 sin (\frac{x}{10})) + C

\int 2 e^{2 x} x d x

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

\int \frac{1}{x ln ( 3 )} d x

\int \frac{10 s + 10}{( s ^{2} + 1 ) ( s - 1 ) ^{3}} d s

x - 1

What is the integral of sin(1/5 x)sin(1/10 x) ?
The integral of sin(1/5 x)sin(1/10 x) is 1/2 (-10/3 sin((3x)/(10))+10sin(x/(10)))+C