"Evaluate the following integral. ∫(3x+4)(3x2+8x−13)5dx" solved and explained

e1s2kat26

Answered question

2021-10-11

Evaluate the following integral.

\int (3 x + 4) {(3 x^{2} + 8 x - 13)}^{5} d x

Answer & Explanation

tabuordy

Skilled2021-10-12Added 90 answers

Step 1
Given integral is
$\int (3 x + 4) {(3 x^{2} + 8 x - 13)}^{5} d x$
We will use the following substitution to evaluate the integral
$3 x^{2} + 8 x - 13 = t$
$(6 x + 8) d x = d t$
$2 (3 x + 4) d x = d t$
$(3 x + 4) d x = \frac{1}{2} d t$
Step 2
Therefore, we have
$\int (3 x + 4) {(3 x^{2} + 8 x - 13)}^{5} d x = \frac{1}{2} \int t^{5} d t$
$\int (3 x + 4) {(3 x^{2} + 8 x - 13)}^{5} d x = \frac{1}{2} (\frac{t^{5 + 1}}{5 + 1}) + c$
$\int (3 x + 4) {(3 x^{2} + 8 x - 13)}^{5} d x = \frac{1}{12} t^{6} + c$
$\int (3 x + 4) {(3 x^{2} + 8 x - 13)}^{5} d x = \frac{1}{12} {(3 x^{2} + 8 x - 13)}^{6} + c$
Step 3
Ans:
$\int (3 x + 4) {(3 x^{2} + 8 x - 13)}^{5} d x = \frac{1}{12} {(3 x^{2} + 8 x - 13)}^{6} + c$

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