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Question

Mathematics Question on Integrals of Some Particular Functions

If dxx+x7=p(x)\int \frac{dx}{x+x^{7}} = p\left(x\right) then, x6x+x7dx\int \frac{x^{6}}{x+x^{7}}dx is equal to:

A

Inxp(x)+cIn \left|x\right| -p\left(x\right) + c

B

Inx+p(x)+cIn \left|x\right| +p\left(x\right) + c

C

xp(x)+cx-p \left(x\right) + c

D

x+p(x)+cx+p \left(x\right) + c

Answer

Inxp(x)+cIn \left|x\right| -p\left(x\right) + c

Explanation

Solution

x6x+x7dx=x6x(1+x6)dx\int \frac{x^{6}}{x+x^{7}}dx = \int \frac{x^{6}}{x\left(1+x^{6}\right)}dx
=(1+x6)1x(1+x6)dx= \int \frac{\left(1+x^{6}\right)-1}{x\left(1+x^{6}\right)}dx
=1xdx1x+x7dx=\int \frac{1}{x}dx-\int \frac{1}{x+x^{7}}dx
=Inxp(x)+c= In \left|x\right| -p\left(x\right) + c