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Question

(x+a)3x3+1x41x7xx21+x+2(x+1)2+(1+x)3xdx is equal to

A
x+3alogx3a2xa32x2+x+x22+x33+x447sec1x+log(x+1)1x+1+2x+2x32+65x52+27x72+C
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B
x+2alogx3a2xa32x2+x+x22+x33+x447sec1x+log(x+1)1x+1+2x+2x32+65x52+27x72+C
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C
x+3alogx3a2xa32x2+x+x22+x33+x447sec1x+log(x+1)1x+1+2x+2x32+65x52+27x52+C
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D
none of these
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Solution

The correct option is A x+3alogx3a2xa32x2+x+x22+x33+x447sec1x+log(x+1)1x+1+2x+2x32+65x52+27x72+C
{(x+a)3x3+1x41x7xx21+x+2(x+1)2+(1+x)3x}dx
.
=x3+3ax2+3a2x+a3x3dx+(1x)(1+x+x2+x3)(1x)dx7xx21dx+(x+1)(x+1)2dx+1(x+1)2dx+1+3x+3x2+x3x12dx
.
Using integration of each function separately, we get
=(x+3alogx3a2xa32x2)+(x+x22+x33+x44)(7sec1x)+(log(x+1))(1x+1)+(2x+2x32+65x52+27x72)+C

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