What is the integral of (ax+b)^4 ?

The integral of (ax+b)^4 is ((ax+b)^5)/(5a)+C

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integral of (ax+b)^4

\int (a x + b)^{4} d x

\frac{( a x + b ) ^{5}}{5 a} + C

Solution steps

\int (a x + b)^{4} d x

Apply u-substitution

: \int \frac{u ^{4}}{a} d u

= \int \frac{u ^{4}}{a} d u

Take the constant out:

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

= \frac{1}{a} \cdot \int u^{4} d u

Apply the Power Rule

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

= \frac{1}{a} \cdot \frac{u ^{5}}{5}

Substitute back

u = a x + b

= \frac{1}{a} \cdot \frac{( a x + b ) ^{5}}{5}

Simplify

\frac{1}{a} \cdot \frac{( a x + b ) ^{5}}{5} : \frac{( a x + b ) ^{5}}{5 a}

= \frac{( a x + b ) ^{5}}{5 a}

Add a constant to the solution

= \frac{( a x + b ) ^{5}}{5 a} + C

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What is the integral of (ax+b)^4 ?
The integral of (ax+b)^4 is ((ax+b)^5)/(5a)+C