alleryStart learning →∫ from 0 to 1 x² dx =
- 1/2
- 1/3
- 1/4
- 1
Show answer and solution
Correct answer: B. 1/3
∫₀¹ x² dx = [x³/3]₀¹ = 1/3 - 0 = 1/3. (Area under parabola from 0 to 1.)
OJEE Integral Calculus — practice questions — page 2
Explore 88 free Integral Calculus questions in OJEE Mathematics. Open each question or exercise to read its content, reference answers and available explanation.
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Showing questions 31–60 of 88.
∫ 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 →Complete the arc length relationRead the answer and worked solution →Match each identity to its nameRead the answer and worked solution →The Pythagorean identitiesRead the answer and worked solution →Odd function or even function?Read the answer and worked solution →Complete the degree and radian conversionRead the answer and worked solution →True or false: identities and parityRead the answer and worked solution →Match each sum or difference formula to its resultRead the answer and worked solution →Signs in the expansionRead the answer and worked solution →Which formula does this belong to?Read the answer and worked solution →True or false: periodicityRead the answer and worked solution →Complete the complementary angle rulesRead the answer and worked solution →Assertion and reasonRead the answer and worked solution →Two are true, one is notRead the answer and worked solution →Three of these are wrong. Find the one that holds.Read the answer and worked solution →
Worked practice questions
∫ ln x dx (by parts: u = ln x, dv = dx):
- ln x / x
- x ln x - x + C
- x ln x
- ln²x
Show answer and solution
Correct answer: B. x ln x - x + C
u = ln x, du = dx/x; dv = dx, v = x.
We have ∫ ln x dx = x ln x - ∫ x × (1/x).
So dx = x ln x - x + C.
∫ from -a to a (odd function) dx =
- f(a) - f(-a)
- 0 (odd function symmetric about origin)
- 2 × ∫ from 0 to a
- ∞
Show answer and solution
Correct answer: B. 0 (odd function symmetric about origin)
For odd function (f(-x) = -f(x)): ∫_(-a)^a f(x) dx = 0. (Negative half cancels positive half.) For even functions (f(-x) = f(x)): ∫_(-a)^a = 2 × ∫_0^a.
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