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Evaluate : `int (2xdx)/((x^(2)+1)^(3//2)` |
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Answer» Method 1 `"Let" " "I=int(2xdx)/(3sqrt(x^(2)+1))` `"Let"u=x^(2)+1, "then" " " du=2xdx`. `I = int u^(-1//3) du` [In the form of ` u^(n)` du] `=(u^(2//3))/(2//3)+C` [Integrate with respect to u] `=(3)/(2) u^(2//3)+C` `(x^(2)+1)^(2//3)+C` [Replace u by ` x^(2)+1]` Method 2 Let ` u=3sqrt(x^(2)+1) rArr u^(3)=x^(2)+1` Then `3u^(2)du=2xdx` `I = int (2xdx)/(3 sqrt(x^(2)+1))` `=int(3u^(2)du)/(u)` `=3.int u du` ` =3.(u^(2))/(2)+C` [ Intergrate with respect to u ] `=(3)/(2) (x^(2)+1)^(2//3)+C` [Replace u by ` (x^(2)+1)^(1//3)`] |
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