Question

score: 0/35 answered: 0/9
question 1
consider the integral below.
$int \frac{4x^6}{(6+x^7)^8}dx$
which integral corresponds to a correct $u$-substitution?
$circ int \frac{9u^{-8}}{28}du$
$circ int \frac{2u^8 - u}{21}du$
$circ int \frac{4}{7u^{-8}}du$
$circ int \frac{4u^{-8}}{7}du$
$circ int 4u^8 du$
$circ$ none of the above

Explanation:

Step1: Choose substitution variable

Let $u = 6 + x^7$

Step2: Compute derivative of u

$\frac{du}{dx} = 7x^6 \implies du = 7x^6 dx \implies x^6 dx = \frac{1}{7}du$

Step3: Rewrite integral in terms of u

Substitute $u = 6 + x^7$ and $x^6 dx = \frac{1}{7}du$ into the original integral:
$\int \frac{4x^6}{(6+x^7)^8}dx = \int \frac{4}{u^8} \cdot \frac{1}{7}du = \int \frac{4u^{-8}}{7}du$

Step4: Match with given options

The rewritten integral matches one of the provided options.

Answer:

$\boldsymbol{\int \frac{4u^{-8}}{7}du}$