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Evaluate `int(2zdz)/(root3(z^2+1))` |
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Answer» Method 1 Let `I=int(2zdz)/(root3(z^2+1))`. Let `u=z^2+1`, then `du=2zdz`. `I=intu^(-1//3)du` [In the form of `u^ndu`] `=(u^(2//3))/(2//3)+C` [Integrate with respect to u] `=3/2u^(2//3)+C` `=(z^2+1)^(2//3)+C` [Replace `u` by `z^2+1`] Method 2 Let `u=root3(z^2+1)impliesu^3=z^2+1` Then `3u^2du=2zdz` `I=int(2zdz)/(root3(z^2+1))` `=int(3u^2du)/(u)` `=3*intudu` `=3*u^2/2+C` [Integrate with respect to u] `=3/2(z^2+1)^(2//3)+C` [Replace u by `(z^2+1)^(1//3)`] |
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