GATE 2026 EE – Question 65

Engineering Mathematics · Calculus: Mean value theorems, Theorems of integral calculus, Evaluation of definite and improper integrals, Partial derivatives, Multiple integrals, Fourier series · 2 marks · Numerical answer

The integral

$$\frac1\pi\int_0^\infty\frac{x^{2026}}{(1+x^{2026})(1+x^2)}dx$$

evaluates to _________.

(Round off to two decimal places)

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Show answer and explanation

Correct answer: 0.24 to 0.26

Explanation

Let $I=\int_0^\infty\dfrac{x^n}{(1+x^n)(1+x^2)}dx$ with $n=2026$. Substituting $x\to1/x$ gives $I=\int_0^\infty\dfrac{1}{(1+x^n)(1+x^2)}dx$. Adding the two forms: $2I=\int_0^\infty\dfrac{1+x^n}{(1+x^n)(1+x^2)}dx=\int_0^\infty\dfrac{dx}{1+x^2}=\dfrac\pi2$. So $I=\dfrac\pi4$ and $\dfrac{I}{\pi}=0.25$.