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

The integral of sqrt(x+1)(x+2) is 2(1/5 (x+1)^{5/2}+1/3 (x+1)^{3/2})+C

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

\int x + 1 (x + 2) d x

2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} + \frac{1}{3} (x + 1)^{\frac{3}{2}}) + C

Solution steps

\int x + 1 (x + 2) d x

Apply u-substitution

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

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

Take the constant out:

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

= 2 \cdot \int u^{2} (u^{2} + 1) d u

Expand

u^{2} (u^{2} + 1) : u^{4} + u^{2}

= 2 \cdot \int u^{4} + u^{2} 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

= 2 (\int u^{4} d u + \int u^{2} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

\int u^{2} d u = \frac{u ^{3}}{3}

= 2 (\frac{u ^{5}}{5} + \frac{u ^{3}}{3})

Substitute back

u = x + 1

= 2 (\frac{( x + 1 ) ^{5}}{5} + \frac{( x + 1 ) ^{3}}{3})

Simplify

2 (\frac{( x + 1 ) ^{5}}{5} + \frac{( x + 1 ) ^{3}}{3}) : 2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} + \frac{1}{3} (x + 1)^{\frac{3}{2}})

= 2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} + \frac{1}{3} (x + 1)^{\frac{3}{2}})

Add a constant to the solution

= 2 (\frac{1}{5} (x + 1)^{\frac{5}{2}} + \frac{1}{3} (x + 1)^{\frac{3}{2}}) + C

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What is the integral of sqrt(x+1)(x+2) ?
The integral of sqrt(x+1)(x+2) is 2(1/5 (x+1)^{5/2}+1/3 (x+1)^{3/2})+C