What is the integral of sin(2x)cos^2(x) ?

The integral of sin(2x)cos^2(x) is -1/2 cos^4(x)+C

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integral of sin(2x)cos^2(x)

\int sin (2 x) cos^{2} (x) d x

- \frac{1}{2} cos^{4} (x) + C

Solution steps

\int sin (2 x) cos^{2} (x) d x

Apply u-substitution

: \int - u d u

= \int - u d u

Take the constant out:

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

= - \int u d u

Apply the Power Rule

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

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

Substitute back

u = cos^{2} (x)

= - \frac{( cos ^{2} ( x ) ) ^{2}}{2}

Simplify

- \frac{( cos ^{2} ( x ) ) ^{2}}{2} : - \frac{1}{2} cos^{4} (x)

= - \frac{1}{2} cos^{4} (x)

Add a constant to the solution

= - \frac{1}{2} cos^{4} (x) + C

\frac{d}{d x} (x^{3} + 2 x^{2} - 1)

\int x (1 - x)^{99} d x

\int \frac{x}{x ^{4} + 4 x ^{2} + 5} d x

x \to 2 + lim (\frac{2 - x}{∣ x - 2 ∣})

t \to 0 lim (\frac{e ^{\frac{1}{t}}}{t})

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