Question

Evaluate each of the following integrals. \int_{0}^{2}(x^{2}+2x-3)^{3}(4x+4)dx

Applications of integrals
ANSWERED
asked 2021-06-03
Evaluate each of the following integrals.
\(\int_{0}^{2}(x^{2}+2x-3)^{3}(4x+4)dx\)

Answers (1)

2021-06-04

Step 1
To evaluate each of the following integrals.
Step 2
Given that
\(\int_{0}^{2}(x^{2}+2x-3)^{3}(4x+4)dx\)
\(=\int_{0}^{2}4(x^{2}+2x-3)^{3}(x+1)dx\)
Let \(x^{2}+2x-3=t\)
Then \(\frac{dt}{dx}=2x+2\)
\(dt=2(x+1)dx\)
Also at \(x=0\) we have\( t=-3\)
At \(x=2\) we have \(t=2^{2}+4-3=8-3=5\)
Hence we have
\(\int_{0}^{2}(x^{2}+2x-3)^{3}(4x+4)dx\)
\(=2\int_{-3}^{5}t^{3}dt\)
\(=2[\frac{t^{4}}{4}]_{-3}^{5}=\frac{1}{2}[5^{4}-(-3)^{4}]\)
\(=\frac{1}{2}[625-81]=272\)
\(\therefore\int_{0}^{2}(x^{2}+2x-3)^{3}(4x+4)dx=272\)

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