Expert Answer to "Determine whether the integral is divergent or convergent...."

geduiwelh

Answered question

2021-09-11

Determine whether the integral is divergent or convergent. If it is convergent, evaluate it. If not, state your answer as "divergent".
$\int_{3}^{\infty} (\frac{8}{{(x + 4)}^{3}} / 2)$

Answer & Explanation

Luvottoq

Skilled2021-09-12Added 95 answers

Compute the definite integral:
$\int_{3}^{\infty} \frac{4}{{(x + 4)}^{3}} d x$
Factor out constants:
$= 4 \int_{3}^{\infty} \frac{1}{{(x + 4)}^{3}} d x$
For the integrand $\frac{1}{{(x + 4)}^{3}}$ , substitute u = x + 4 and du = dx.
This gives a new lower bound $u = 4 + 3 = 7$ and upper bound $u = \infty$ :
$= 4 \int_{7}^{\infty} \frac{1}{u^{3}} d u$
Apply the fundamental theorem of calculus.
The antiderivative of $\frac{1}{u^{3}} i s - \frac{1}{2 u^{2}}$ :
$= lim_{b \to \infty} (- \frac{2}{u^{2}}) |_{7}^{b}$
Evaluate the antiderivative at the limits and subtract.
$(- \frac{2}{u^{2}}) |_{7}^{\infty} = (- \frac{2}{\infty^{2}}) - (- \frac{2}{7^{2}}) = \frac{2}{49}$ :
Answer:
$= \frac{2}{49}$

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