Question

Integration Procedures (Integration by Parts)

∫ y √(y+1)dy

Answer 1

So, the solution is (integration by parts):

$\int y \sqrt{y+1} dy = \int y \sqrt{y+1} d(y+1) = \\ \int y d(\frac{2}{3}(y+1)^{\frac{3}{2}}) = \frac{2}{3}y(y+1)^{\frac{3}{2}} - \\ - \int \frac{2}{3}(y+1)^{\frac{3}{2}} d(y+1) = \frac{2}{3}y(y+1)^{\frac{3}{2}} - \\ - \frac{2}{3} \frac{2}{5} (y+1)^{\frac{5}{2}} + const = \frac{2}{3}(y+1)^{\frac{3}{2}} \cdot \\ \cdot [y - \frac{2}{5}(y+1) +const = \frac{2}{15}(y+1)^{\frac{3}{2}}[3y-2] + \\ + const$