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. \( -\frac{1}{2} \ln \left|1+4 x^{2}\right|-4 \ln |x+2|+C \)

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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. \( \frac{1}{4} \ln \left|1+4 x^{2}\right|+2 \ln |x+2|+C \)
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