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

The integral of (x+3)(x+1)^{1/2} is 2/3 x(x+1)^{3/2}-4/15 (x+1)^{5/2}+2(x+1)^{3/2}+C

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integral of (x+3)(x+1)^{1/2}

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

\frac{2}{3} x (x + 1)^{\frac{3}{2}} - \frac{4}{15} (x + 1)^{\frac{5}{2}} + 2 (x + 1)^{\frac{3}{2}} + C

Solution steps

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

Expand

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

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

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

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

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

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

Add a constant to the solution

= \frac{2}{3} x (x + 1)^{\frac{3}{2}} - \frac{4}{15} (x + 1)^{\frac{5}{2}} + 2 (x + 1)^{\frac{3}{2}} + C

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What is the integral of (x+3)(x+1)^{1/2} ?
The integral of (x+3)(x+1)^{1/2} is 2/3 x(x+1)^{3/2}-4/15 (x+1)^{5/2}+2(x+1)^{3/2}+C