What is the integral of 2ysin(x)cos(x) ?

The integral of 2ysin(x)cos(x) is ysin^2(x)+C

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integral of 2ysin(x)cos(x)

\int 2 y sin (x) cos (x) d x

y sin^{2} (x) + C

Solution steps

\int 2 y sin (x) cos (x) d x

Take the constant out:

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

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

Apply u-substitution

: \int u d u

= 2 y \cdot \int u d u

Apply the Power Rule

: \frac{u ^{2}}{2}

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

Substitute back

u = sin (x)

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

Simplify

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

= y sin^{2} (x)

Add a constant to the solution

= y sin^{2} (x) + C

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What is the integral of 2ysin(x)cos(x) ?
The integral of 2ysin(x)cos(x) is ysin^2(x)+C