IIT-JEE 2010 · Paper 1 · Mathematics Q41

MathematicsMultiple correct+3Question 41 of 84

The value(s) of $\displaystyle\int_0^1\frac{x^4(1-x)^4}{1+x^2}\,dx$ is (are)

\dfrac{22}7-\pi

\dfrac2{105}

\dfrac{71}{15}-\dfrac{3\pi}2

Answer and worked solution

Official answer: A

Polynomial division.
$\frac{x^4(1-x)^4}{1+x^2}=x^6-4x^5+5x^4-4x^2+4-\frac4{1+x^2}.$

Integrate on [0, 1].
$\frac17-\frac46+1-\frac43+4-4\cdot\frac\pi4=\frac{22}7-\pi.$
(The integrand is positive, which proves $\pi<\frac{22}7$ .)

(A)

Topics:Properties of definite integrals Partial fractions

IIT-JEE 2010 · Paper 1, Mathematics question 41