What is the integral of (4x+3)sqrt(1+2x) ?

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

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integral of (4x+3)sqrt(1+2x)

\int (4 x + 3) 1 + 2 x d x

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

Solution steps

\int (4 x + 3) 1 + 2 x d x

Apply Integration By Parts

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

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

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

Add a constant to the solution

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

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