What is the integral of 3(x-2)^4 ?

The integral of 3(x-2)^4 is 3/5 (x-2)^5+C

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integral of 3(x-2)^4

\int 3 (x - 2)^{4} d x

\frac{3}{5} (x - 2)^{5} + C

Solution steps

\int 3 (x - 2)^{4} d x

Take the constant out:

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

= 3 \cdot \int (x - 2)^{4} d x

Apply u-substitution

: \int u^{4} d u

= 3 \cdot \int u^{4} d u

Apply the Power Rule

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

= 3 \cdot \frac{u ^{5}}{5}

Substitute back

u = x - 2

= 3 \cdot \frac{( x - 2 ) ^{5}}{5}

Simplify

3 \cdot \frac{( x - 2 ) ^{5}}{5} : \frac{3}{5} (x - 2)^{5}

= \frac{3}{5} (x - 2)^{5}

Add a constant to the solution

= \frac{3}{5} (x - 2)^{5} + C

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