"Evaluate the integrals. ∫(3x+1)(3x2+2x+1)3dx" was answered by our community

waigaK

Answered question

2021-11-05

Evaluate the integrals.

\int (3 x + 1) {(3 x^{2} + 2 x + 1)}^{3} d x

Nola Robson

Skilled2021-11-06Added 94 answers

Step 1
Consider the following integral:

I = \int (3 x + 1) {(3 x^{2} + 2 x + 1)}^{3} d x

Substitute

3 x^{2} + 2 x + 1 = t \to (6 x + 2) d x = d t \to (3 x + 1) d x = \frac{1}{2} d t

in I:

I = \frac{1}{2} \int t^{3} d t

= \frac{1}{2} \times \frac{t^{3 + 1}}{3 + 1} + C

= \frac{1}{2} \times \frac{t^{4}}{4} + C

= \frac{t^{4}}{8} + C

Step 2
Resubstitute

t = 3 x^{2} + 2 x + 1

in I:

I = \frac{{(3 x^{2} + 2 x + 1)}^{4}}{8} + C

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