Question

3.4 homework - substitution method of integration
score: 30/80 answered: 3/8
question 4
evaluate the integral
\\(\int \frac{dx}{(4x + 13)^2}\\),
by making the appropriate substitution: \\(u = \\)
\\(\int \frac{dx}{(4x + 13)^2}\\)
question help: video

Explanation:

Step1: Choose substitution \( u \)

Let \( u = 4x + 13 \). Then, find \( du \) by differentiating \( u \) with respect to \( x \).
\( \frac{du}{dx} = 4 \), so \( du = 4dx \), which implies \( dx = \frac{du}{4} \).

Step2: Substitute into the integral

Substitute \( u = 4x + 13 \) and \( dx = \frac{du}{4} \) into the integral \( \int \frac{dx}{(4x + 13)^2} \).
We get \( \int \frac{\frac{du}{4}}{u^2} = \frac{1}{4} \int u^{-2} du \).

Step3: Integrate \( u^{-2} \)

Using the power rule for integration \( \int x^n dx = \frac{x^{n + 1}}{n + 1} + C \) (for \( n
eq - 1 \)), here \( n=-2 \).
\( \frac{1}{4} \int u^{-2} du=\frac{1}{4}\times\frac{u^{-2 + 1}}{-2 + 1}+C=\frac{1}{4}\times\frac{u^{-1}}{-1}+C=-\frac{1}{4u}+C \).

Step4: Substitute back \( u = 4x + 13 \)

Substitute \( u = 4x + 13 \) into the result: \( -\frac{1}{4(4x + 13)}+C \).

Answer:

For the substitution, \( u = \boldsymbol{4x + 13} \).
The integral \( \int \frac{dx}{(4x + 13)^2} \) evaluates to \( \boldsymbol{-\frac{1}{4(4x + 13)} + C} \) (where \( C \) is the constant of integration).