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

The integral of sin(1-x)(2-cos(1-x))^4 is -((2-cos(1-x))^5)/5+C

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integral of sin(1-x)(2-cos(1-x))^4

\int sin (1 - x) (2 - cos (1 - x))^{4} d x

- \frac{( 2 - cos ( 1 - x ) ) ^{5}}{5} + C

Solution steps

\int sin (1 - x) (2 - cos (1 - x))^{4} d x

Apply u-substitution

: \int - u^{4} d u

= \int - u^{4} d u

Take the constant out:

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

= - \int u^{4} d u

Apply the Power Rule

: \frac{u ^{5}}{5}

= - \frac{u ^{5}}{5}

Substitute back

u = 2 - cos (1 - x)

= - \frac{( 2 - cos ( 1 - x ) ) ^{5}}{5}

Add a constant to the solution

= - \frac{( 2 - cos ( 1 - x ) ) ^{5}}{5} + C

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What is the integral of sin(1-x)(2-cos(1-x))^4 ?
The integral of sin(1-x)(2-cos(1-x))^4 is -((2-cos(1-x))^5)/5+C