For every whole number n ≥ 0 let In = ∫₀¹ xⁿ eˣ dx. So…

Quizzes
For every whole number n ≥ 0 let In = ∫₀¹ xⁿ eˣ dx. So I₀ = e - 1. Integration by parts, with xⁿ as the part to differentiate, links each In to the one before it. Then the values can be found one after another without integrating again, starting from I₀. Some of the statements below follow from that link; others need a comparison of the integrands, and one is about the size of In for large n. Decide about each statement on its own. Which of the following statements are true?

MathematicsAdvancedHigh School (11–12)Functions and Calculus › Integrals JEE Advanced

For every whole number $n \ge 0$ let $\displaystyle I_n = \int_0^1 x^n e^x\,dx.$ So $I_0 = e - 1$ . Integration by parts, with $x^n$ as the part to differentiate, links each $I_n$ to the one before it. Then the values can be found one after another without integrating again, starting from $I_0$ . Some of the statements below follow from that link; others need a comparison of the integrands, and one is about the size of $I_n$ for large $n$ . Decide about each statement on its own. Which of the following statements are true?

One or more options are correct. Tick all the correct ones: a wrong option costs marks, and a part-right answer earns part of the marks.

← All Quizzes

← Each entry of List-I is a definite integral or a limit of sums, and each has an exact value. Find each one, and match it with its value in List-II. Some entries of List-II may be used more than once, and some not at all. Each entry of List-I needs a different idea: a limit of sums, a symmetry of the interval, integration by parts, and a square root that hides a modulus. List-I P. lim_(n → ∞) Σ_(k=1)ⁿ 1/(√(n² + nk)) Q. ∫₀^(π/2) sin 2x ln(tan x) dx R. ∫₀¹ tan⁻¹ x dx S. ∫₀^(π/2) √(1 - sin 2x) dx List-II 1. 0 2. 2(√2 - 1) 3. √2 - 1 4. π/4 - 12 ln 2 5. π/4 + 12 ln 2 Let F(x) = ∫₀^(x²) sin√ t dt for every real x. The upper limit x² is the same for x and for -x, so F is an even function. Near t = 0 the integrand behaves like √ t, and that decides how fast F(x) goes to 0. Recall that √(x²) = |x|, not x, which matters as soon as x is negative. Two of the statements are about what happens very close to x = 0, from one side or from both sides, and one is about the sign of F far from 0. The substitution t = w² gives F in closed form, but most of the statements can be decided without it. Which of the following statements are true? →

More onFunctions and Calculus › Integrals

Discussion (0)

Loading…

Discussion (0)

Problems on this topic