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∫ ( 132(x)(x - 1)^10 dx )

Factor out 132.

132 ∫ ( (x)(x - 1)^10 dx )

To solve this, use substitution.

Let u = x - 1. Then

u + 1 = x, so

du = dx

132 ∫ ( (u + 1)u^(10) du )

Distribute u^10 over the brackets.

132 ∫ ( ( u^11 + u^10 ) du )

Integrate using the power rule.

132 [ (1/12)u^(12) + (1/11)u^(11) ] + C

Distribute the 132.

(132/12)u^(12) + (132/11)u^(11) + C

Reduce each fraction.

11u^(12) + 12u^(11) + C

But u = x - 1, so the final answer is

11[x - 1]^(12) + 12[x - 1]^(11) + C

Let's factor (x - 1)^(11) out of this.

(x - 1)^11 (11(x - 1) + 12) + C

Reducing

(x - 1)^11 (11x - 11 + 12) + C

(x - 1)^11 (11x + 1) + C

By parts = 132( 1/11* (x-1)^11* x -1/11 Int(x-1)^11 dx)=

132( 1/11*x *(x-1)^11-1/132 (x-1)^12 +C =

=(x-1)^11(12x -x+1)=(x-1)^11(11x+1)+C Answer(c)

i got c, use u substitution u = x-1

the answer is c