What is the integral of (x^2)/(sqrt(x^2+2x+10)) ?

The integral of (x^2)/(sqrt(x^2+2x+10)) is

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integral of (x^2)/(sqrt(x^2+2x+10))

\int \frac{x ^{2}}{x ^{2} + 2 x + 10} d x

9 (- \frac{1}{2} ln (\frac{x + 1 + x ^{2} + 2 x + 10}{3}) + \frac{1}{18} (x + 1) x^{2} + 2 x + 10) - 2 x^{2} + 2 x + 10 + ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10) + C

Solution steps

\int \frac{x ^{2}}{x ^{2} + 2 x + 10} d x

Complete the square

x^{2} + 2 x + 10 : (x + 1)^{2} + 9

= \int \frac{x ^{2}}{( x + 1 ) ^{2} + 9} d x

Apply u-substitution

: \int \frac{( u - 1 ) ^{2}}{u ^{2} + 9} d u

= \int \frac{( u - 1 ) ^{2}}{u ^{2} + 9} d u

Expand

\frac{( u - 1 ) ^{2}}{u ^{2} + 9} : \frac{u ^{2}}{u ^{2} + 9} - \frac{2 u}{u ^{2} + 9} + \frac{1}{u ^{2} + 9}

= \int \frac{u ^{2}}{u ^{2} + 9} - \frac{2 u}{u ^{2} + 9} + \frac{1}{u ^{2} + 9} 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

= \int \frac{u ^{2}}{u ^{2} + 9} d u - \int \frac{2 u}{u ^{2} + 9} d u + \int \frac{1}{u ^{2} + 9} d u

\int \frac{u ^{2}}{u ^{2} + 9} d u = 9 (- \frac{1}{2} ln (\frac{u + 9 + u ^{2}}{3}) + \frac{1}{18} u 9 + u^{2})

\int \frac{2 u}{u ^{2} + 9} d u = 2 u^{2} + 9

\int \frac{1}{u ^{2} + 9} d u = ln (\frac{1}{3} u + 9 + u^{2})

= 9 (- \frac{1}{2} ln (\frac{u + 9 + u ^{2}}{3}) + \frac{1}{18} u 9 + u^{2}) - 2 u^{2} + 9 + ln (\frac{1}{3} u + 9 + u^{2})

Substitute back

u = x + 1

= 9 - \frac{1}{2} ln \frac{x + 1 + 9 + ( x + 1 ) ^{2}}{3} + \frac{1}{18} (x + 1) 9 + (x + 1)^{2} - 2 (x + 1)^{2} + 9 + ln (\frac{1}{3} x + 1 + 9 + (x + 1)^{2})

Simplify

9 - \frac{1}{2} ln \frac{x + 1 + 9 + ( x + 1 ) ^{2}}{3} + \frac{1}{18} (x + 1) 9 + (x + 1)^{2} - 2 (x + 1)^{2} + 9 + ln (\frac{1}{3} x + 1 + 9 + (x + 1)^{2}) : 9 (- \frac{1}{2} ln (\frac{x + 1 + x ^{2} + 2 x + 10}{3}) + \frac{1}{18} (x + 1) x^{2} + 2 x + 10) - 2 x^{2} + 2 x + 10 + ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10)

= 9 (- \frac{1}{2} ln (\frac{x + 1 + x ^{2} + 2 x + 10}{3}) + \frac{1}{18} (x + 1) x^{2} + 2 x + 10) - 2 x^{2} + 2 x + 10 + ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10)

Add a constant to the solution

= 9 (- \frac{1}{2} ln (\frac{x + 1 + x ^{2} + 2 x + 10}{3}) + \frac{1}{18} (x + 1) x^{2} + 2 x + 10) - 2 x^{2} + 2 x + 10 + ln (\frac{1}{3} x + 1 + x^{2} + 2 x + 10) + C

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What is the integral of (x^2)/(sqrt(x^2+2x+10)) ?
The integral of (x^2)/(sqrt(x^2+2x+10)) is