Question

\(\displaystyle \int \frac{3x^{2}+2}{x^{3}+2x}\,dx=\ \ ?\)

• A. \(\sin^{-1}(x^{3}+3x)+k\)
• B. \(\tan^{-1}(3x^{2}+2)+k\)
• C. \(\log|3x^{2}+2|+k\)
• D. \(\log|x^{3}+2x|+k\)

Hint:

Try to see the denominator's derivative sitting upstairs.

Answer 1

The Correct Option is D

Let \(g(x)=x^{3}+2x\Rightarrow g'(x)=3x^{2}+2\) (the numerator). Therefore \(\int\dfrac{g'(x)}{g(x)}dx=\ln|g(x)|+k=\ln|x^{3}+2x|+k\).