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

The integral of (2x+5)(x+1)^{1/2} is 2/3 (2x+5)(x+1)^{3/2}-8/15 (x+1)^{5/2}+C

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integral of (2x+5)(x+1)^{1/2}

\int (2 x + 5) (x + 1)^{\frac{1}{2}} d x

\frac{2}{3} (2 x + 5) (x + 1)^{\frac{3}{2}} - \frac{8}{15} (x + 1)^{\frac{5}{2}} + C

Solution steps

\int (2 x + 5) (x + 1)^{\frac{1}{2}} d x

Apply Integration By Parts

: \frac{2}{3} (2 x + 5) (x + 1)^{\frac{3}{2}} - \int \frac{4}{3} (x + 1)^{\frac{3}{2}} d x

= \frac{2}{3} (2 x + 5) (x + 1)^{\frac{3}{2}} - \int \frac{4}{3} (x + 1)^{\frac{3}{2}} d x

\int \frac{4}{3} (x + 1)^{\frac{3}{2}} d x = \frac{8}{15} (x + 1)^{\frac{5}{2}}

= \frac{2}{3} (2 x + 5) (x + 1)^{\frac{3}{2}} - \frac{8}{15} (x + 1)^{\frac{5}{2}}

Add a constant to the solution

= \frac{2}{3} (2 x + 5) (x + 1)^{\frac{3}{2}} - \frac{8}{15} (x + 1)^{\frac{5}{2}} + C

e^{- 2 x} sin (x), 0

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

\frac{d}{d x} (\frac{6 x ^{2}}{( x ^{3} + 1 ) ^{2}})

y^{^{''}} + 3 y^{^{'}} = 0, y (0) = 2, y^{^{'}} (0) = - 3

\int \frac{1}{1 - 3 y} d y

What is the integral of (2x+5)(x+1)^{1/2} ?
The integral of (2x+5)(x+1)^{1/2} is 2/3 (2x+5)(x+1)^{3/2}-8/15 (x+1)^{5/2}+C