Ford Lawson
12/06/2023 · Senior High School
Evaluate the integral using partial fractions \( I=\int \frac{10 x^{2}+4 x+2}{\left(1+4 x^{2}\right)(x+2)} d x \) A. \( \frac{1}{4} \ln \left|1+4 x^{2}\right|+2 \ln |x+2|+C \) B. \( -\frac{1}{4} \ln \left|1+4 x^{2}\right|+\ln |x+2|+C \) C. \( \frac{1}{2} \ln \left|1+4 x^{2}\right|+4 \ln |x+2|+C \) D. None of the options E. \( -\frac{1}{2} \ln \left|1+4 x^{2}\right|-4 \ln |x+2|+C \)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
Calculate the integral \( \int \frac{10x^2+4x+2}{(1+4x^2)(x+2)}dx \).
Evaluate the integral by following steps:
- step0: Evaluate:
\(\int \frac{10x^{2}+4x+2}{\left(1+4x^{2}\right)\left(x+2\right)} dx\)
- step1: Multiply the terms:
\(\int \frac{10x^{2}+4x+2}{x+2+4x^{3}+8x^{2}} dx\)
- step2: Rewrite the fraction:
\(\int \frac{2}{2+x}+\frac{2x}{1+4x^{2}} dx\)
- step3: Use properties of integrals:
\(\int \frac{2}{2+x} dx+\int \frac{2x}{1+4x^{2}} dx\)
- step4: Evaluate the integral:
\(2\ln{\left(\left|x+2\right|\right)}+\frac{1}{4}\ln{\left(4x^{2}+1\right)}\)
- step5: Use the logarithm base change rule:
\(\ln{\left(\left(x+2\right)^{2}\right)}+\ln{\left(\left(4x^{2}+1\right)^{\frac{1}{4}}\right)}\)
- step6: Transform the expression:
\(\ln{\left(\left(x+2\right)^{2}\left(4x^{2}+1\right)^{\frac{1}{4}}\right)}\)
- step7: Add the constant of integral C:
\(\ln{\left(\left(x+2\right)^{2}\left(4x^{2}+1\right)^{\frac{1}{4}}\right)} + C, C \in \mathbb{R}\)
The integral \( I=\int \frac{10 x^{2}+4 x+2}{\left(1+4 x^{2}\right)(x+2)} d x \) evaluates to \( \ln{((x+2)^{2}(4x^{2}+1)^{\frac{1}{4}})} + C \), where \( C \) is an arbitrary constant.
Therefore, the correct answer is:
A. \( \frac{1}{4} \ln \left|1+4 x^{2}\right|+2 \ln |x+2|+C \)
Quick Answer
The correct answer is A. \( \frac{1}{4} \ln \left|1+4 x^{2}\right|+2 \ln |x+2|+C \)
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit