Step by step solution to "Use the Change of Variables Formula to evaluate..."

Burhan Hopper

Answered question

2021-10-29

Use the Change of Variables Formula to evaluate the definite integral.

\int_{0}^{1} (x + 1) {(x^{2} + 2 x)}^{5} d x

Talisha

Skilled2021-10-30Added 93 answers

Step 1
Given integral:

\int_{0}^{1} (x + 1) {(x^{2} + 2 x)}^{5} d x

Now,
Substitute:

x^{2} + 2 x = u

Differentiate both sides with respect to x:

\frac{d (x^{2} + 2 x)}{d x} = \frac{d u}{d x}

\Rightarrow 2 x + 2 = \frac{d u}{d x}

\Rightarrow (x + 1) d x = \frac{d u}{2}

Also,
At

x = 0 \Rightarrow u = 0^{2} + 2 \times 0 = 0

x = 1 \Rightarrow u = 1^{2} + 2 \times 1 = 3

Step 2
Therefore,
Integral become:

\int_{0}^{1} (x + 1) {(x^{2} + 2 x)}^{5} d x = \int_{0}^{3} {(u)}^{5} \frac{d u}{2}

= \frac{1}{2} {[\frac{u^{5 + 1}}{5 + 1}]}_{0}^{3}

= \frac{1}{2} {[\frac{u^{6}}{6}]}_{0}^{3}

= \frac{1}{2} [\frac{3^{6}}{6} - \frac{0^{6}}{6}]

= \frac{3^{6}}{12}

= \frac{243}{4}

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