What is the integral of (ax+b)^{3/2} ?

The integral of (ax+b)^{3/2} is (2(ax+b)^{5/2})/(5a)+C

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integral of (ax+b)^{3/2}

\int (a x + b)^{\frac{3}{2}} d x

\frac{2 ( a x + b ) ^{\frac{5}{2}}}{5 a} + C

Solution steps

\int (a x + b)^{\frac{3}{2}} d x

Apply u-substitution

= \int \frac{u ^{\frac{3}{2}}}{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^{\frac{3}{2}} d u

Apply the Power Rule

= \frac{1}{a} \cdot \frac{2}{5} u^{\frac{5}{2}}

Substitute back

u = a x + b

= \frac{1}{a} \cdot \frac{2}{5} (a x + b)^{\frac{5}{2}}

Simplify

\frac{1}{a} \cdot \frac{2}{5} (a x + b)^{\frac{5}{2}} : \frac{2 ( a x + b ) ^{\frac{5}{2}}}{5 a}

= \frac{2 ( a x + b ) ^{\frac{5}{2}}}{5 a}

Add a constant to the solution

= \frac{2 ( a x + b ) ^{\frac{5}{2}}}{5 a} + C

What is the integral of (ax+b)^{3/2} ?
The integral of (ax+b)^{3/2} is (2(ax+b)^{5/2})/(5a)+C