Question

question 4
0 pts
your mathematical work must accurately support your final answer. correct answers receive no credit without
mathematically correct supporting work.
(10 points) evaluate by method of substitution: $\int \frac{4 x-6}{\left(x^{2}-3 x+8\
ight)^{10}} d x$
o click to proceed when you have completed the work on your own paper.

Explanation:

Step1: Let \( u = x^{2}-3x + 8 \)

Differentiate \( u \) with respect to \( x \): \( du=(2x - 3)dx \). Notice that \( 4x-6 = 2(2x - 3) \). So, \( (4x - 6)dx=2du \)

Step2: Substitute into the integral

The integral \( \int\frac{4x - 6}{(x^{2}-3x + 8)^{10}}dx=\int\frac{2du}{u^{10}} \)

Step3: Integrate \( \frac{2}{u^{10}} \)

Using the power - rule for integration \( \int u^{n}du=\frac{u^{n + 1}}{n+1}+C\) (\(n
eq - 1\)), for \(n=-10\), we have \( \int\frac{2}{u^{10}}du=2\int u^{-10}du\)

$$
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Step4: Substitute back \( u=x^{2}-3x + 8 \)

The result is \( -\frac{2}{9(x^{2}-3x + 8)^{9}}+C \)

Answer:

\( -\frac{2}{9(x^{2}-3x + 8)^{9}}+C \)