Question

homework 2: 1.5 substitution
score: 14/35 answered: 5/9
question 6
consider the indefinite integral $int x^{6} sqrt5{x^{7}+7} dx$.
a) this can be transformed using the substitution
$u = $
which gives $du = $ (dont forget the differential $dx$ or $du$.)
c) performing the substitution in terms of $u$ gives the integral
$int $ .(dont forget the differential.)

Explanation:

Step1: Choose substitution variable

Let $u = x^7 + 7$

Step2: Compute derivative of u

$du = 7x^6 dx$

Step3: Isolate $x^6 dx$

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

Step4: Substitute into integral

Rewrite $\sqrt[5]{x^7+7}$ as $u^{1/5}$, then substitute:
$\int u^{1/5} \cdot \frac{1}{7} du = \frac{1}{7} \int u^{1/5} du$

Answer:

a) $u = x^7 + 7$
$du = 7x^6 dx$
c) $\frac{1}{7} u^{1/5} du$ (or $\frac{1}{7} \sqrt[5]{u} du$)