Exams › JEE Advanced › Maths

Evaluate the integral: integral of e^(sin x) * (5 cos x + cos³ x) / ((2 - sin x) * sqrt(3 + cos² x)) dx

1. e^(sin x) * 1/sqrt(4 - sin² x) + C
2. e^(sin x) * 1/sqrt(2 - cos x) + C
3. e^(sin x) * sqrt((2 + sin x)/(2 - sin x)) + C
4. e^(sin x) * 1/sqrt(2 + cos x) + C

Correct answer: e^(sin x) * sqrt((2 + sin x)/(2 - sin x)) + C

Solution

Letting t=sin x converts the integral to e^t*(6-t²)/((2-t)^(3/2)*sqrt(2+t)) dt. Writing g(t)=sqrt((2+t)/(2-t)), one can show g(t)+g'(t) equals exactly that expression, so the antiderivative is e^t*g(t) = e^(sin x)*sqrt((2+sin x)/(2-sin x)) + C.

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 →