What is the integral of x/(sqrt(x-7)) ?

The integral of x/(sqrt(x-7)) is 2(1/3 (x-7)^{3/2}+7sqrt(x-7))+C

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integral of x/(sqrt(x-7))

\int \frac{x}{x - 7} d x

2 (\frac{1}{3} (x - 7)^{\frac{3}{2}} + 7 x - 7) + C

Solution steps

\int \frac{x}{x - 7} d x

Apply u-substitution

: \int 2 (u^{2} + 7) d u

= \int 2 (u^{2} + 7) d u

Take the constant out:

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

= 2 \cdot \int u^{2} + 7 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

= 2 (\int u^{2} d u + \int 7 d u)

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

\int 7 d u = 7 u

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

Substitute back

u = x - 7

= 2 (\frac{( x - 7 ) ^{3}}{3} + 7 x - 7)

Simplify

2 (\frac{( x - 7 ) ^{3}}{3} + 7 x - 7) : 2 (\frac{1}{3} (x - 7)^{\frac{3}{2}} + 7 x - 7)

= 2 (\frac{1}{3} (x - 7)^{\frac{3}{2}} + 7 x - 7)

Add a constant to the solution

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

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What is the integral of x/(sqrt(x-7)) ?
The integral of x/(sqrt(x-7)) is 2(1/3 (x-7)^{3/2}+7sqrt(x-7))+C