What is the integral of (x^2)/(sqrt(x^2)+21) ?

The integral of (x^2)/(sqrt(x^2)+21) is ((x+21)^2)/2-42(x+21)+441ln|x+21|+C

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integral of (x^2)/(sqrt(x^2)+21)

integral

\frac{x ^{2}}{x ^{2} + 21}

\frac{( x + 21 ) ^{2}}{2} - 42 (x + 21) + 441 ln ∣ x + 21 ∣ + C

Solution steps

\int \frac{x ^{2}}{x ^{2} + 21} d x

x^{2} = (x),

assuming

x \geq 0

= \int \frac{x ^{2}}{x + 21} d x

Apply u-substitution

= \int \frac{( u - 21 ) ^{2}}{u} d u

Expand

\frac{( u - 21 ) ^{2}}{u} : u - 42 + \frac{441}{u}

= \int u - 42 + \frac{441}{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

= \int u d u - \int 42 d u + \int \frac{441}{u} d u

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

\int 42 d u = 42 u

\int \frac{441}{u} d u = 441 ln ∣ u ∣

= \frac{u ^{2}}{2} - 42 u + 441 ln ∣ u ∣

Substitute back

u = x + 21

= \frac{( x + 21 ) ^{2}}{2} - 42 (x + 21) + 441 ln ∣ x + 21 ∣

Add a constant to the solution

= \frac{( x + 21 ) ^{2}}{2} - 42 (x + 21) + 441 ln ∣ x + 21 ∣ + C

What is the integral of (x^2)/(sqrt(x^2)+21) ?
The integral of (x^2)/(sqrt(x^2)+21) is ((x+21)^2)/2-42(x+21)+441ln|x+21|+C