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

The integral of (x^2+1)sqrt(2x+1) is 1/7 x^2(2x+1)^{3/2}-2/35 x(2x+1)^{3/2}+37/105 (2x+1)^{3/2}+C

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

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

\frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{37}{105} (2 x + 1)^{\frac{3}{2}} + C

Solution steps

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

Apply Integration By Parts

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

\int \frac{2}{3} x (2 x + 1)^{\frac{3}{2}} d x = \frac{2}{105} (10 x^{2} (2 x + 1)^{\frac{3}{2}} + 3 x (2 x + 1)^{\frac{3}{2}} - (2 x + 1)^{\frac{3}{2}})

= \frac{1}{3} (x^{2} + 1) (2 x + 1)^{\frac{3}{2}} - \frac{2}{105} (10 x^{2} (2 x + 1)^{\frac{3}{2}} + 3 x (2 x + 1)^{\frac{3}{2}} - (2 x + 1)^{\frac{3}{2}})

Simplify

\frac{1}{3} (x^{2} + 1) (2 x + 1)^{\frac{3}{2}} - \frac{2}{105} (10 x^{2} (2 x + 1)^{\frac{3}{2}} + 3 x (2 x + 1)^{\frac{3}{2}} - (2 x + 1)^{\frac{3}{2}}) : \frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{37}{105} (2 x + 1)^{\frac{3}{2}}

= \frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{37}{105} (2 x + 1)^{\frac{3}{2}}

Add a constant to the solution

= \frac{1}{7} x^{2} (2 x + 1)^{\frac{3}{2}} - \frac{2}{35} x (2 x + 1)^{\frac{3}{2}} + \frac{37}{105} (2 x + 1)^{\frac{3}{2}} + C

in v erse l a pl a ce \frac{( 3 s e ^{(- 2 s)} )}{4 s ^{2} + 36}

\int x^{2} cos (x) - 2 x cos (x) d x

x \to 9 lim (\frac{x - 9}{∣ x - 9 ∣})

x \to 0 lim (ln (x))

in v erse l a pl a ce \frac{5}{( s ^{3} - 10 \cdot s ^{2} )}

What is the integral of (x^2+1)sqrt(2x+1) ?
The integral of (x^2+1)sqrt(2x+1) is 1/7 x^2(2x+1)^{3/2}-2/35 x(2x+1)^{3/2}+37/105 (2x+1)^{3/2}+C