What is the integral of xsqrt(ax+b) ?

The integral of xsqrt(ax+b) is (2(5ax(ax+b)^{3/2}-2(ax+b)^{5/2}))/(15a^2)+C

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integral of xsqrt(ax+b)

\int x a x + b d x

\frac{2 ( 5 a x ( a x + b ) ^{\frac{3}{2}} - 2 ( a x + b ) ^{\frac{5}{2}} )}{15 a ^{2}} + C

Solution steps

\int x a x + b d x

Apply u-substitution

= \int \frac{2 u ^{2} ( u ^{2} - b )}{a ^{2}} d u

Take the constant out:

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

= \frac{2}{a ^{2}} \cdot \int u^{2} (u^{2} - b) d u

Apply Integration By Parts

= \frac{2}{a ^{2}} (\frac{1}{3} u^{3} (u^{2} - b) - \int \frac{2 u ^{4}}{3} d u)

\int \frac{2 u ^{4}}{3} d u = \frac{2 u ^{5}}{15}

= \frac{2}{a ^{2}} (\frac{1}{3} u^{3} (u^{2} - b) - \frac{2 u ^{5}}{15})

Substitute back

u = a x + b

= \frac{2}{a ^{2}} (\frac{1}{3} (a x + b)^{3} ((a x + b)^{2} - b) - \frac{2 ( a x + b ) ^{5}}{15})

Simplify

\frac{2}{a ^{2}} (\frac{1}{3} (a x + b)^{3} ((a x + b)^{2} - b) - \frac{2 ( a x + b ) ^{5}}{15}) : \frac{2 ( 5 a x ( a x + b ) ^{\frac{3}{2}} - 2 ( a x + b ) ^{\frac{5}{2}} )}{15 a ^{2}}

= \frac{2 ( 5 a x ( a x + b ) ^{\frac{3}{2}} - 2 ( a x + b ) ^{\frac{5}{2}} )}{15 a ^{2}}

Add a constant to the solution

= \frac{2 ( 5 a x ( a x + b ) ^{\frac{3}{2}} - 2 ( a x + b ) ^{\frac{5}{2}} )}{15 a ^{2}} + C

What is the integral of xsqrt(ax+b) ?
The integral of xsqrt(ax+b) is (2(5ax(ax+b)^{3/2}-2(ax+b)^{5/2}))/(15a^2)+C