alleryIndefinite integral of xⁿ (n ≠ -1):
- xⁿ⁺¹/(n+1) + C
- xⁿ⁻¹/n
- n xⁿ⁻¹
- xⁿ
Show answer and solution
Correct answer: A. xⁿ⁺¹/(n+1) + C
∫ xⁿ dx = xⁿ⁺¹/(n+1) + C (n ≠ -1).
Reverse of power rule for differentiation.
For n = -1: ∫ x⁻¹ dx = ln|x| + C.
JEE Main Unit 6B: Definite Integration & Riemann Sums — practice questions
Explore 27 free Unit 6B: Definite Integration & Riemann Sums questions in JEE Main Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
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Indefinite integral of xⁿ (n ≠ -1):Read the answer and worked solution →∫ (1/x) dx =Read the answer and worked solution →∫ e^x dx =Read the answer and worked solution →∫ sin x dx =Read the answer and worked solution →∫ cos x dx =Read the answer and worked solution →Linearity of integration: ∫ [af(x) + bg(x)] dx =Read the answer and worked solution →∫ tan x dx =Read the answer and worked solution →Integration by substitution: ∫ 2x × e^(x²) dx = (let u = x²)Read the answer and worked solution →∫ from 0 to π/2 sin x dx =Read the answer and worked solution →Integration by parts formula:Read the answer and worked solution →∫ x e^x dx (by parts: u = x, dv = e^x dx):Read the answer and worked solution →∫ from 0 to 1 x² dx =Read the answer and worked solution →∫ ln x dx (by parts: u = ln x, dv = dx):Read the answer and worked solution →∫ from -a to a (odd function) dx =Read the answer and worked solution →Fundamental theorem of calculus: ∫ from a to b f'(x) dx =Read the answer and worked solution →∫ 1/(1 + x²) dx =Read the answer and worked solution →∫ from 0 to π sin²x dx =Read the answer and worked solution →∫ x sin x dx (integration by parts):Read the answer and worked solution →∫ sec²x dx =Read the answer and worked solution →Area under y = x² from x = 0 to x = 2:Read the answer and worked solution →∫ from 0 to ∞ e^(-x) dx =Read the answer and worked solution →∫ x ln x dx (integration by parts: u = ln x, dv = x dx):Read the answer and worked solution →∫ from 1 to e (1/x) dx =Read the answer and worked solution →Reduction formula example: ∫ sinⁿ x dx (for n ≥ 2) =Read the answer and worked solution →∫ from 0 to 2 (x² - 2x) dx =Read the answer and worked solution →For ∫ dx / sqrt(a² - x²), let x = a sin θ. Result:Read the answer and worked solution →∫ from 0 to π/2 sin x cos x dx =Read the answer and worked solution →
Worked practice questions
∫ (1/x) dx =
- x
- x²
- -1/x²
- ln|x| + C
Show answer and solution
Correct answer: D. ln|x| + C
∫ dx/x = ln|x| + C. Special case of power rule that doesn't apply (n = -1).
Absolute value ensures domain validity for both positive and negative x.
∫ e^x dx =
- e^x + C
- x e^x
- ln x
- x²
Show answer and solution
Correct answer: A. e^x + C
∫ e^x dx = e^x + C. (Exponential is its own integral up to constant.) Similarly ∫ e^(kx) dx = e^(kx)/k + C.
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