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Questions  >>  JEEMAIN and NEET  >>  Mathematics  >>  Class12  >>  Integral Calculus
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Q)

Integrate : $\int \large\frac{x^5+1}{x^2+1}$$dx$

\[\begin {array} {1 1} (a)\;\frac{x^4}{4}-\frac{x^2}{2}+\frac{1}{2} \log (x^2+1)+\tan ^{-1} (x^2+1) +c \\ (b)\;\frac{x^4}{4}+\frac{x^2}{2}+\frac{1}{2} \log (x^2+1)+\tan ^{-1} (x^2-1) +c \\ (c)\;x^2+x+c \\ (d)\;None \end {array}\]

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A)
$\int \bigg(x^3-x+\large\frac{x+1}{x^2+1}\bigg)$$dx$
$\int (x^3-x) dx +\int \large\frac{x+1}{x^3+1}$$dx$
$x^2 +1 =t$
$2xdx =dt$
$\int (x^3-x) dx +\int \large\frac{dt}{2(t)}+\int \large\frac{1}{x^2+1}$$dx$
=>$ \large\frac{x^4}{4}-\frac{x^2}{2}+\frac{1}{2}$$ \log (t)+\tan ^{-1}(x)+c$
=>$ \large\frac{x^4}{4}-\frac{x^2}{2}+\frac{1}{2}$$ \log (x^2+1)+\tan ^{-1} (x^2+1) +c$
Hence a is the correct answer.
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