What is the integral of (x^2)/(sqrt(x-25)) ?

The integral of (x^2)/(sqrt(x-25)) is 2(1/5 (x-25)^{5/2}+50/3 (x-25)^{3/2}+625sqrt(x-25))+C

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integral of (x^2)/(sqrt(x-25))

\int \frac{x ^{2}}{x - 25} d x

2 (\frac{1}{5} (x - 25)^{\frac{5}{2}} + \frac{50}{3} (x - 25)^{\frac{3}{2}} + 625 x - 25) + C

Solution steps

\int \frac{x ^{2}}{x - 25} d x

Apply u-substitution

: \int 2 (u^{2} + 25)^{2} d u

= \int 2 (u^{2} + 25)^{2} 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} + 25)^{2} d u

Expand

(u^{2} + 25)^{2} : u^{4} + 50 u^{2} + 625

= 2 \cdot \int u^{4} + 50 u^{2} + 625 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^{4} d u + \int 50 u^{2} d u + \int 625 d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int 50 u^{2} d u = \frac{50 u ^{3}}{3}

\int 625 d u = 625 u

= 2 (\frac{u ^{5}}{5} + \frac{50 u ^{3}}{3} + 625 u)

Substitute back

u = x - 25

= 2 (\frac{( x - 25 ) ^{5}}{5} + \frac{50 ( x - 25 ) ^{3}}{3} + 625 x - 25)

Simplify

2 (\frac{( x - 25 ) ^{5}}{5} + \frac{50 ( x - 25 ) ^{3}}{3} + 625 x - 25) : 2 (\frac{1}{5} (x - 25)^{\frac{5}{2}} + \frac{50}{3} (x - 25)^{\frac{3}{2}} + 625 x - 25)

= 2 (\frac{1}{5} (x - 25)^{\frac{5}{2}} + \frac{50}{3} (x - 25)^{\frac{3}{2}} + 625 x - 25)

Add a constant to the solution

= 2 (\frac{1}{5} (x - 25)^{\frac{5}{2}} + \frac{50}{3} (x - 25)^{\frac{3}{2}} + 625 x - 25) + C

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What is the integral of (x^2)/(sqrt(x-25)) ?
The integral of (x^2)/(sqrt(x-25)) is 2(1/5 (x-25)^{5/2}+50/3 (x-25)^{3/2}+625sqrt(x-25))+C