A) \[\frac{{{(x+2)}^{10}}}{10}-\frac{{{(x+2)}^{8}}}{8}+C\]
B) \[\frac{{{(x+1)}^{2}}}{2}-\frac{{{(x+2)}^{8}}}{8}-\frac{{{(x+3)}^{2}}}{2}+C\]
C) \[\frac{{{(x+2)}^{10}}}{10}+C\]
D) \[\frac{{{(x+1)}^{2}}}{2}+\frac{{{(x+2)}^{8}}}{8}+\frac{{{(x+3)}^{2}}}{2}+C\]
E) \[\frac{{{(x+2)}^{9}}}{9}-\frac{{{(x+2)}^{7}}}{7}+C\]
Correct Answer: A
Solution :
Let\[I=\int{(x+1){{(x+2)}^{7}}(x+3)}dx\] Putting \[x+2=t\] \[\Rightarrow \] \[dx=dt\] Also,\[x+1=t-1\]and \[x+3=t+1\] \[\therefore \] \[I=\int{(t-1){{t}^{7}}}(t+1)dt\] \[=\int{({{t}^{2}}-1)}{{t}^{7}}dt\] \[=\int{{{t}^{9}}}dt-\int{{{t}^{7}}}dt\] \[=\frac{{{t}^{10}}}{10}-\frac{{{t}^{8}}}{8}+c\] \[=\frac{{{(x+2)}^{10}}}{10}-\frac{{{(x+2)}^{8}}}{8}+c\]
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