What is the integral of cos^4(5-x)sin(5-x) ?

The integral of cos^4(5-x)sin(5-x) is 1/5 cos^5(5-x)+C

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integral of cos^4(5-x)sin(5-x)

\int cos^{4} (5 - x) sin (5 - x) d x

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

Solution steps

\int cos^{4} (5 - x) sin (5 - x) d x

Apply u-substitution

: \int - cos^{4} (u) sin (u) d u

= \int - cos^{4} (u) sin (u) d u

Take the constant out:

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

= - \int cos^{4} (u) sin (u) d u

Apply u-substitution

: \int - v^{4} d v

= - \int - v^{4} d v

Take the constant out:

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

= - (- \int v^{4} d v)

Apply the Power Rule

: \frac{v ^{5}}{5}

= - (- \frac{v ^{5}}{5})

Substitute back

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

Simplify

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

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

Add a constant to the solution

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

\frac{d y}{d x} = \frac{3 + y}{1 - 2 x}, y (0) = 0

y^{^{''}} - 3 y^{^{'}} = - 6 x + 4

x^{2} - 4 x, x = 0

\frac{d}{d x} (\frac{A + B x + C x ^{2}}{x ^{2}})

\frac{d}{d x} (x e^{x y})

What is the integral of cos^4(5-x)sin(5-x) ?
The integral of cos^4(5-x)sin(5-x) is 1/5 cos^5(5-x)+C