Q. If $\int_{}^{}{\frac{1}{x + x^{5}}dx = f(x) + c}$, then the value of $\int_{}^{}{\frac{x^{4}}{x + x^{5}}\ dx}$ equals
A) $\log{x - f(x) + c}$
B) $f(x) + \log{x + c}$
C) $f(x) - \log{x + c}$
D) None of these
Explanation
(a)
Let $I = \int_{}^{}{\frac{x^{4}}{x + x^{5}}\ dx} = \int_{}^{}{\frac{x^{4} + 1 - 1}{x + x^{5}}\ dx}$
$\Rightarrow \ I = \int_{}^{}{\frac{1}{x}\ dx} - \int_{}^{}{\frac{1}{x^{5} + x}\ dx}$
$\Rightarrow \ I = \log{x - \left\lbrack f(x) + c \right\rbrack}$ [given]
$\Rightarrow \ I = \log x - f(x) + c_{1}$ [where $c_{1} = - c$]
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.