What is the integral of (x(x+2))/(sqrt(x^3+3x^2+1)) ?

The integral of (x(x+2))/(sqrt(x^3+3x^2+1)) is 2/3 sqrt(x^3+3x^2+1)+C

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integral of (x(x+2))/(sqrt(x^3+3x^2+1))

\int \frac{x ( x + 2 )}{x ^{3} + 3 x ^{2} + 1} d x

\frac{2}{3} x^{3} + 3 x^{2} + 1 + C

Solution steps

\int \frac{x ( x + 2 )}{x ^{3} + 3 x ^{2} + 1} d x

Apply u-substitution

: \int \frac{1}{3 u} d u

= \int \frac{1}{3 u} d u

Take the constant out:

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

= \frac{1}{3} \cdot \int \frac{1}{u} d u

Apply the Power Rule

: 2 u^{\frac{1}{2}}

= \frac{1}{3} \cdot 2 u^{\frac{1}{2}}

Substitute back

u = x^{3} + 3 x^{2} + 1

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

Simplify

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

= \frac{2}{3} x^{3} + 3 x^{2} + 1

Add a constant to the solution

= \frac{2}{3} x^{3} + 3 x^{2} + 1 + C

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