Answer to Question #178350 in Calculus for Phyroe

Question #178350

Integration Procedures (Integration by Parts)


∫ y √(y+1)dy


1
Expert's answer
2021-04-25T07:22:33-0400

So, the solution is (integration by parts):


yy+1dy=yy+1d(y+1)=yd(23(y+1)32)=23y(y+1)3223(y+1)32d(y+1)=23y(y+1)322325(y+1)52+const=23(y+1)32[y25(y+1)+const=215(y+1)32[3y2]++const\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




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