Exams › JEE Advanced › Maths

Evaluate the integral: integral from 0 to infinity of x / [(1 + x)(1 + x²)] dx. Which of the following is true?

1. The integral equals pi / 4
2. The integral equals pi / 2
3. The integral equals integral from 0 to infinity of 1 / [(1 + x)(1 + x²)] dx
4. The integral cannot be evaluated

Correct answer: The integral equals pi / 4

Solution

Partial fractions: x/[(1+x)(1+x²)] = A/(1+x) + (Bx+C)/(1+x²). Multiplying through: x = A(1+x²) + (Bx+C)(1+x). Comparing coefficients: x²: 0 = A+B => B = -A. x¹: 1 = B+C => C = 1-B = 1+A. x⁰: 0 = A+C => C = -A. So 1+A = -A => A = -1/2, B = 1/2, C = 1/2. Integral = -1/2 * integral(1/(1+x))dx + 1/2 * integral(x/(1+x²))dx + 1/2 * integral(1/(1+x²))dx from 0 to inf. = -1/2*[ln(1+x)]₀^inf + 1/4*[ln(1+x²)]₀^inf + 1/2*[arctan(x)]₀^inf. The ln terms: -1/2*ln(1+x) + 1/4*ln(1+x²) = -1/2*ln(1+x) + 1/2*ln(sqrt(1+x²)). As x->inf: -1/2*ln(x) + 1/2*ln(x) = 0. So log terms cancel and the result is 1/2*(pi/2 - 0) = pi/4.

Related JEE Advanced Maths questions

• Let f(x) = 4x / (4x² + 2). If the sum of the integrals ∫(1/1997) + ∫(2/1997) +... + ∫(1196/1997) equals 499q, what is the value of q?
• Evaluate the integral ∫₀¹⁰⁰π [(arccot x) + (arctan x)] dx, which equals 100π + cot p. Determine the value of p, where [.] represents the greatest integer function.
• If the integral ∫₀² f(x) dx = -4, determine the value of d such that ∫₀² (-2d²) dx = -4.
• The derivative of y with respect to x is given as dy/dx = 3x² - 4x + 1. After integration, y becomes x³ - 2x² + x + B. If y equals 5 when x is 1, what is the value of B?
• Evaluate the integral ∫_(π/4)^(π/2) (2 csc x)¹⁷ dx. Which of the following expressions represents its value?
• The derivative of a function is given by f'(x) = (192x³)/(2 + sin⁴(π x)) for all x ∈ R, and it is known that f ((1)/(2)) = 0. If the integral ∫_(1/2)¹ f(x) dx is bounded such that m ≤ ∫_(1/2)¹ f(x) dx ≤ M, what are the possible values of m and M?

⚔️ Practice JEE Advanced Maths free + battle 1v1 →