A) \[\frac{{{(x+1)}^{7/2}}}{7}-2\frac{{{(x+1)}^{5/2}}}{5}\]\[+2\frac{{{(x+1)}^{3/2}}}{3}+C\]
B) \[2\left[ \frac{{{(x+1)}^{7/2}}}{7}-2\frac{{{(x+1)}^{5/2}}}{5} \right.\]\[\left. +2\frac{{{(x+1)}^{3/2}}}{3} \right]+c\]
C) \[\frac{{{(x+1)}^{7/2}}}{7}-2\frac{{{(x+1)}^{5/2}}}{5}+c\]
D) \[\frac{{{(x+1)}^{7/2}}}{7}-3\frac{{{(x+1)}^{5/2}}}{5}\]\[+11{{(x+1)}^{1/2}}+c\]
E) \[{{(x+1)}^{7/2}}+{{(x+1)}^{5/2}}+{{(x+1)}^{3/2}}+c\]
Correct Answer: B
Solution :
Let \[I=\int{({{x}^{2}}+1)}\sqrt{x+1}dx\] \[=\int{{{x}^{2}}}\sqrt{x+1}dx+\int{\sqrt{x+1}}dx\] Putting \[x+1=r\] \[\Rightarrow \] \[dx=dt\] Also, \[x=t-1\] \[\therefore \] \[z=\int{{{(t-1)}^{2}}}\sqrt{t}dt+\int{\sqrt{t}}dt\] \[=\int{({{t}^{5/2}}-2{{t}^{3/2}}+{{t}^{1/2}})}dt+\int{\sqrt{t}}dt\] \[=\frac{2}{7}{{t}^{7/2}}-\frac{4}{5}{{t}^{5/2}}+\frac{2}{3}{{t}^{3/2}}+\frac{2}{3}{{t}^{3/2}}+c\] \[=2\left( \frac{{{(x+1)}^{7/2}}}{7}-\frac{2{{(x+1)}^{5/2}}}{4}+\frac{2}{3}{{(x+1)}^{3/2}} \right)+c\]
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