Question

What will be the integration of the following wrt x?

${\int }_0^{x}x{\left(1+x\right)}^n$

Answer 1

$$\begin{gathered} Dearstudent \\ I={\int }_0^{x}x{\left(1+x\right)}^{n}dx \\ Let1+x=t \\ x=t-1 \\ dt=dx \\ I={\int }_1^{x+1}\left(t-1\right)t^{n}dt={\int }_1^{x+1}{t}^{n+1}dt-{\int }_1^{x+1}{t}^{n}dt \\ I={\left[\frac{t^{n+2}}{n+2}\right]}_1^{x+1}-{\left[\frac{t^{n+1}}{n+1}\right]}_1^{x+1} \\ =\frac{{\left(x+1\right)}^{n+2}}{n+2}-\frac{1}{n+2}-\frac{{\left(x+1\right)}^{n+1}}{n+1}+\frac{1}{n+1} \\ =\frac{{\left(x+1\right)}^{n+2}}{n+2}-\frac{{\left(x+1\right)}^{n+1}}{n+1}+\frac{1}{\left(n+1\right)\left(n+1\right)} \end{gathered}$$

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