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

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

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

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

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

Solution steps

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

Apply u-substitution

: \int u^{2} d u

= \int u^{2} d u

Apply the Power Rule

: \frac{u ^{3}}{3}

= \frac{u ^{3}}{3}

Substitute back

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

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

Simplify

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

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

Add a constant to the solution

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

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