Question

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

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

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
$$\int_{0}^{2}(x^{2}+2x-3)^{3}(4x+4)dx$$
$$=2\int_{-3}^{5}t^{3}dt$$
$$=2[\frac{t^{4}}{4}]_{-3}^{5}=\frac{1}{2}[5^{4}-(-3)^{4}]$$
$$=\frac{1}{2}[625-81]=272$$
$$\therefore\int_{0}^{2}(x^{2}+2x-3)^{3}(4x+4)dx=272$$