First slide

Introduction to integration

Question

$\int \frac{1}{x + x^{5}} d x = f (x) + c, t h e n t h e v a l u e o f \int \frac{x^{4}}{x + x^{5}} d x =$

Easy

A

$\log x - f (x) + c$
B

$f (x) + \log x + c$
C

$f (x) - \log x + c$
D

None of these

Solution

$\begin{array}{l} \int \frac{x^{4}}{x + x^{5}} d x ( a d d & s u b t r a c t 1 i n t h e n u m e r a t o r) \\ = \int \frac{x^{4} + 1}{x + x^{5}} d x - \int \frac{1}{x + x^{5}} d x \\ = \int \frac{1 + x^{4}}{x (1 + x^{4})} d x - \int \frac{1}{x + x^{5}} d x \\ = \int \frac{1}{x} d x - \int \frac{1}{x + x^{5}} d x \\ = \log x - f (x) + c (g i v e n \int \frac{1}{x + x^{5}} d x = f (x)) \end{array}$

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