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

The integral of sin(2x)tan(x) is x-1/2 sin(2x)+C

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integral of sin(2x)tan(x)

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

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

Solution steps

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

Rewrite using trig identities

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

Take the constant out:

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

= 2 \cdot \int cos (x) sin (x) tan (x) d x

Rewrite using trig identities

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

Take the constant out:

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

= 2 \cdot \frac{1}{2} \cdot \int 1 - cos (2 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

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

\int 1 d x = x

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

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

Simplify

2 \cdot \frac{1}{2} (x - \frac{1}{2} sin (2 x)) : x - \frac{1}{2} sin (2 x)

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

Add a constant to the solution

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

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