Question

Evaluate: $\int \frac{ x^{3}}{(x+1)^{2}}\ dx$

Answer 1

The answer is provided in the attachment.
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Answer 2

Answer:

$\int\:\frac{x^{3} }{(x+1)^{2} }dx= \frac{(x+1)^{2} }{2}-3(x+1)+3\:log\:|x+1|}+\frac{1}{x+1}+C$

Step-by-step explanation:

we can perform this integration by substitution method

let

$x+1=t\\\\dx = dt\\\\and\\\\x=t-1\\\\x^{3} =(t-1)^{3} \\\\and\\\\(t-1)^{3}=t^{3} -1-3t^{2}+3t\\ \\=t^{3} -3t^{2}+3t-1$

now substitute this value

$\int\:\frac{x^{3} }{(x+1)^{2} }dx\\\\\int\:\frac{(t-1)^{3} }{t^{2} }dt=\int\:\frac{t^{3} -3t^{2}+3t-1}{t^{2}}$

$=\int\:(t-3+\frac{3}{t}-\frac{1}{t^{2} })dt\\ \\ = \int\:t\:dt-\int\:3\:dt+\int\:\frac{3}{t}\:dt-\int\:\frac{1}{t^{2}}\:dt\\ \\ \\=\frac{t^{2} }{2}-3t+3\:log\:t+\frac{1}{t}+C$

Now undo substitution

$\int\:\frac{x^{3} }{(x+1)^{2} }dx= \frac{(x+1)^{2} }{2}-3(x+1)+3\:log\:|x+1|}+\frac{1}{x+1}+C$