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Question

Mathematics Question on integral

If ∫1x+x5dx=f(x)+c,\int \frac{1}{x+x^{5}}dx =f\left(x\right)+c , then ∫x4x+x5dx=\int \frac{x^{4}}{x+x^{5}} dx =

A

log⁡∣x∣+f(x)+c\log | x | + f(x) + c

B

log⁡∣x∣−f(x)+c\log | x | - f(x) + c

C

log⁡∣x∣+xf(x)+c\log | x | + x f(x) + c

D

log⁡∣x∣−xf(x)+c\log | x | - x f(x) + c

Answer

log⁡∣x∣−f(x)+c\log | x | - f(x) + c

Explanation

Solution

∫1x+x5dx=f(x)+c\int \frac{1}{x+x^{5}}dx =f\left(x\right)+c
I=∫x4x+x5dx=∫x4+1−1x(1+x4)dxI = \int \frac{x^{4}}{x+x^{5}} dx = \int \frac{x^{4} +1 -1}{x \left(1+x^{4}\right)} dx
=∫x4+1x(1+x4)dx−∫1x(1+x4)dx= \int \frac{x^4+1}{x(1+x^4)} dx - \int \frac{1}{x(1+x^4)} dx
=∫1xdx−∫1x+x5dx= \int \frac{1}{x} dx - \int \frac{1}{x+x^5} dx
=log∣x∣−f(x)+c= log |x| - f(x) + c