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

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

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

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

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

Solution steps

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

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

Simplify

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

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

Add a constant to the solution

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

\frac{d}{d x} (x^{- \frac{2}{5}} + x^{\frac{2}{5}})

- \frac{48}{s ^{5}}

\int_{0}^{3} π (2 7 y)^{2} d y

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

y^{^{''}} - 0.2 y^{^{'}} + 0.01 y = 0, y (0) = 13, y^{^{'}} (0) = b

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