What is the integral of xsqrt(29x-7) ?

The integral of xsqrt(29x-7) is 1/841 (2/5 (29x-7)^{5/2}+14/3 (29x-7)^{3/2})+C

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integral of xsqrt(29x-7)

\int x 29 x - 7 d x

\frac{1}{841} (\frac{2}{5} (29 x - 7)^{\frac{5}{2}} + \frac{14}{3} (29 x - 7)^{\frac{3}{2}}) + C

Solution steps

\int x 29 x - 7 d x

Apply u-substitution

: \int \frac{u ( u + 7 )}{841} d u

= \int \frac{u ( u + 7 )}{841} d u

Take the constant out:

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

= \frac{1}{841} \cdot \int u (u + 7) d u

Expand

u (u + 7) : u^{\frac{3}{2}} + 7 u

= \frac{1}{841} \cdot \int u^{\frac{3}{2}} + 7 u d u

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \frac{1}{841} (\int u^{\frac{3}{2}} d u + \int 7 u d u)

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

\int 7 u d u = \frac{14}{3} u^{\frac{3}{2}}

= \frac{1}{841} (\frac{2}{5} u^{\frac{5}{2}} + \frac{14}{3} u^{\frac{3}{2}})

Substitute back

u = 29 x - 7

= \frac{1}{841} (\frac{2}{5} (29 x - 7)^{\frac{5}{2}} + \frac{14}{3} (29 x - 7)^{\frac{3}{2}})

Add a constant to the solution

= \frac{1}{841} (\frac{2}{5} (29 x - 7)^{\frac{5}{2}} + \frac{14}{3} (29 x - 7)^{\frac{3}{2}}) + C

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What is the integral of xsqrt(29x-7) ?
The integral of xsqrt(29x-7) is 1/841 (2/5 (29x-7)^{5/2}+14/3 (29x-7)^{3/2})+C