Question

homework 2: 1.5 substitution
score: 18/35 answered: 6/9
question 7
consider the definite integral $int_{2}^{6} \frac{x^{2}}{(x^{3}+4)^{14}}dx$.
a) this can be transformed using the substitution
$u = square$.
b) this gives $du = square$ (dont forget the differential $dx$ or $du$.)
c) performing the substitution in terms of $u$ and changing the limits of integration gives the integral
$int_{square}^{square} square$.

Explanation:

Step1: Choose substitution variable

Select $u$ as the inner function:
$u = x^3 + 4$

Step2: Find $du$ via differentiation

Differentiate $u$ with respect to $x$:
$\frac{du}{dx} = 3x^2 \implies du = 3x^2 dx$

Step3: Adjust for $x^2 dx$

Isolate $x^2 dx$ for substitution:
$x^2 dx = \frac{1}{3} du$

Step4: Compute new limits for $u$

For $x=2$: $u = 2^3 + 4 = 12$
For $x=6$: $u = 6^3 + 4 = 220$

Step5: Rewrite integral in terms of $u$

Substitute $u$, $du$, and new limits:
$\int_{12}^{220} \frac{1}{3u^{14}} du$

Answer:

a) $x^3 + 4$
b) $3x^2 dx$
c) Lower limit: $12$, Integrand: $\frac{1}{3u^{14}}$, Upper limit: $220$
Transformed integral: $\boldsymbol{\int_{12}^{220} \frac{1}{3u^{14}} du}$