If 7x^8+8x^7/(1+x+x^8)²\,dx=f(x)+c, then f(x) is equal to

Correct Answer is : x81+x+x8x^8/1+x+x^81+x+x8x8

Correct Answer is : x81+x+x8x^8/1+x+x^81+x+x8x8

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EAMCET Engineering 2010 Solved Paper

Section: Mathematics

Question : 43 of 160

Marks: +1, -0

\int\frac{7x^8+8x^7}{(1+x+x^8)^2}\,dx=f(x)+c

f(x)

Solution:

\int\frac{7x^8+8x^7}{(1+x+x^8)^2}\,dx=f(x)+c

here better way to solve this integration we will check the options ie,

we will differentiate the option one by one and check the integration function

Taking option (a) we see that,

f(x)=\frac{d}{dx}\left\{\frac{x^8}{1+x+x^8}\right\}

=\frac{[(1+x+x^8)\cdot8x^7-x^8(1+8x^7)]}{(1+x+x^8)^2}

f(x)=\frac{8x^7+8x^8+8x^{15}-x^8-8x^{15}}{(1+x+x^8)^2}

f(x)=\frac{7x^8+8x^7}{(1+x+x^8)^2}

Hence,

\int\frac{7x^8+8x^7}{(1+x+x^8)^2}\,dx=\frac{x^8}{1+x+x^8}+c

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