What is the integral of xsqrt(a+bx) ?

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

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integral of xsqrt(a+bx)

\int x a + b x d x

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

Solution steps

\int x a + b x d x

Apply u-substitution

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

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

Take the constant out:

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

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

Apply Integration By Parts

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

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

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

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

Substitute back

u = a + b x

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

Simplify

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

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

Add a constant to the solution

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

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