What is the integral of-3sin(5x)cos^2(5x) ?

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

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integral of-3sin(5x)cos^2(5x)

\int - 3 sin (5 x) cos^{2} (5 x) d x

\frac{1}{5} cos^{3} (5 x) + C

Solution steps

\int - 3 sin (5 x) cos^{2} (5 x) d x

Take the constant out:

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

= - 3 \cdot \int sin (5 x) cos^{2} (5 x) d x

Apply u-substitution

: \int sin (u) cos^{2} (u) \frac{1}{5} d u

= - 3 \cdot \int sin (u) cos^{2} (u) \frac{1}{5} d u

Take the constant out:

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

= - 3 \cdot \frac{1}{5} \cdot \int sin (u) cos^{2} (u) d u

Apply u-substitution

: \int - v^{2} d v

= - 3 \cdot \frac{1}{5} \cdot \int - v^{2} d v

Take the constant out:

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

= - 3 \cdot \frac{1}{5} (- \int v^{2} d v)

Apply the Power Rule

: \frac{v ^{3}}{3}

= - 3 \cdot \frac{1}{5} (- \frac{v ^{3}}{3})

Substitute back

= - 3 \cdot \frac{1}{5} (- \frac{cos ^{3} ( 5 x )}{3})

Simplify

- 3 \cdot \frac{1}{5} (- \frac{cos ^{3} ( 5 x )}{3}) : \frac{1}{5} cos^{3} (5 x)

= \frac{1}{5} cos^{3} (5 x)

Add a constant to the solution

= \frac{1}{5} cos^{3} (5 x) + C

x \to 0 lim (5 x)

f (x) = \frac{- 5}{x - 1}, at a = 8

r = 1 + cos (θ)

g (x) = (2 x^{2} + x + 9)^{- 3}

\frac{d}{d y} (e^{3 x} (3 x^{2} y + 2 x y + y^{3}))

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