What is the integral of sin(x)f(cos(x)) ?

The integral of sin(x)f(cos(x)) is f/2 sin^2(x)+C

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integral of sin(x)f(cos(x))

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

\frac{f}{2} sin^{2} (x) + C

Solution steps

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

Take the constant out:

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

= f \cdot \int cos (x) sin (x) d x

Apply u-substitution

= f \cdot \int u d u

Apply the Power Rule

= f \frac{u ^{2}}{2}

Substitute back

u = sin (x)

= f \frac{sin ^{2} ( x )}{2}

Simplify

f \frac{sin ^{2} ( x )}{2} : \frac{f}{2} sin^{2} (x)

= \frac{f}{2} sin^{2} (x)

Add a constant to the solution

= \frac{f}{2} sin^{2} (x) + C

What is the integral of sin(x)f(cos(x)) ?
The integral of sin(x)f(cos(x)) is f/2 sin^2(x)+C