Power Rule & Standard Integrals — JEE Main Basic Maths MCQs with Solutions
Free JEE Main Basic Maths Power Rule & Standard Integrals MCQs with step-by-step solutions (25 questions). Part of Integration. Practise online on Prepizo — no login needed.
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Questions with solutions
Q1 — Power Rule & Standard Integrals · easy · theory
The power rule for integration states $\int x^n\,dx =$
A. $\dfrac{x^{n+1}}{n+1} + C$ (for $n \neq -1$) ✓ Correct
B. $x^{n+1} + C$
C. $n x^{n-1} + C$
D. $\dfrac{x^{n-1}}{n-1} + C$
Solution: Raise the power by one and divide by the new power: ∫xⁿ dx = x^{n+1}/(n+1) + C, valid for all n except n = −1.
Q2 — Power Rule & Standard Integrals · easy · theory
The power rule fails for n = −1. Instead, $\int \dfrac{1}{x}\,dx =$
A. $\ln x^2 + C$
B. $\ln|x| + C$ ✓ Correct
C. $-x^{-2} + C$
D. $\dfrac{x^0}{0} + C$
Solution: For n = −1 the formula would divide by zero; the correct result is ∫(1/x)dx = ln|x| + C.
Q3 — Power Rule & Standard Integrals · easy · theory
The constant of integration C is added because:
A. It makes the integral larger
B. Any constant differentiates to zero, so antiderivatives are unique only up to a constant ✓ Correct
C. It represents the area
D. It is required only for definite integrals
Solution: Since d/dx(C) = 0, F(x) + C is also an antiderivative for any constant C. Forgetting C is the most common indefinite-integral error.
Q4 — Power Rule & Standard Integrals · easy · theory
The linearity property of integration states $\int [a\,f(x) + b\,g(x)]\,dx =$
A. $a\int f(x)\,dx + b\int g(x)\,dx$ ✓ Correct
B. $\int f(x)\,dx + \int g(x)\,dx$ only
C. $\int f(x)\,dx \cdot \int g(x)\,dx$
D. $ab\int f(x)g(x)\,dx$
Solution: Integration is linear: constants factor out and sums integrate term by term.
Q5 — Power Rule & Standard Integrals · easy · theory
Integration is best described as the:
A. Multiplication of functions
B. Process of finding a slope
C. Reverse process of differentiation (antiderivative) ✓ Correct
D. Same as differentiation
Solution: If d/dx F(x) = f(x), then ∫f(x)dx = F(x) + C — integration undoes differentiation.
Q6 — Power Rule & Standard Integrals · easy · theory
The integral of a constant, $\int k\,dx$, equals:
A. $kx + C$ ✓ Correct
B. $\dfrac{k^2}{2} + C$
C. $k + C$
D. $0$
Solution: A constant k = kx⁰, so ∫k dx = kx + C.
Q7 — Power Rule & Standard Integrals · easy · theory
To integrate $\sqrt{x}$, one first rewrites it as:
A. $x^{-1/2}$
B. $x^2$
C. $\ln x$
D. $x^{1/2}$, then applies the power rule ✓ Correct
Solution: √x = x^{1/2}; ∫x^{1/2}dx = x^{3/2}/(3/2) + C = (2/3)x^{3/2} + C.
Q8 — Power Rule & Standard Integrals · medium · theory
The relationship $\dfrac{d}{dx}\left[\int f(x)\,dx\right] = f(x)$ expresses that:
A. Differentiation and integration are inverse operations ✓ Correct
B. C must be zero
C. The integral of a derivative is always zero
D. f(x) must be a polynomial
Solution: Differentiating an antiderivative returns the original integrand — the two operations are inverses.
Q9 — Power Rule & Standard Integrals · medium · theory
Which integral CANNOT be evaluated by the simple power rule?
A. $\int \dfrac{1}{x}\,dx$ ✓ Correct
B. $\int x^2\,dx$
C. $\int x^{-3}\,dx$
D. $\int \sqrt{x}\,dx$
Solution: ∫x⁻¹dx is the n = −1 exception (gives ln|x|); the others are all standard power-rule cases.
Q10 — Power Rule & Standard Integrals · easy · theory
The integral $\int (3x^2 - 4x + 5)\,dx$ is found by:
A. Multiplying the terms first
B. Applying the quotient rule
C. Differentiating each term
D. Integrating each term separately using linearity ✓ Correct
Solution: By linearity, integrate term by term: x³ − 2x² + 5x + C.
Q11 — Power Rule & Standard Integrals · easy · numerical
$\int x^3\,dx =$
A. $4x^4 + C$
B. $3x^2 + C$
C. $\dfrac{x^4}{3} + C$
D. $\dfrac{x^4}{4} + C$ ✓ Correct
Solution: ∫x³dx = x⁴/4 + C.
Q12 — Power Rule & Standard Integrals · easy · numerical
$\int x^5\,dx =$
A. $\dfrac{x^6}{6} + C$ ✓ Correct
B. $\dfrac{x^4}{4} + C$
C. $5x^4 + C$
D. $\dfrac{x^6}{5} + C$
Solution: ∫x⁵dx = x⁶/6 + C.
Q13 — Power Rule & Standard Integrals · easy · numerical
$\int (2x + 3)\,dx =$
A. $x^2 + 3 + C$
B. $2x^2 + 3x + C$
C. $x^2 + 3x + C$ ✓ Correct
D. $2 + C$
Solution: ∫2x dx + ∫3 dx = x² + 3x + C.
Q14 — Power Rule & Standard Integrals · easy · numerical
$\int \dfrac{1}{x^2}\,dx =$
A. $-\dfrac{2}{x^3} + C$
B. $\ln x^2 + C$
C. $\dfrac{1}{x} + C$
D. $-\dfrac{1}{x} + C$ ✓ Correct
Solution: ∫x⁻²dx = x⁻¹/(−1) + C = −1/x + C.
Q15 — Power Rule & Standard Integrals · easy · numerical
$\int \sqrt{x}\,dx =$
A. $\dfrac{1}{2}x^{-1/2} + C$
B. $\dfrac{3}{2}x^{3/2} + C$
C. $2x^{1/2} + C$
D. $\dfrac{2}{3}x^{3/2} + C$ ✓ Correct
Solution: ∫x^{1/2}dx = x^{3/2}/(3/2) + C = (2/3)x^{3/2} + C.
Q16 — Power Rule & Standard Integrals · easy · numerical
$\int (x^2 - 3x + 2)\,dx =$
A. $\dfrac{x^3}{3} - 3x^2 + 2x + C$
B. $2x - 3 + C$
C. $\dfrac{x^3}{3} - \dfrac{3x^2}{2} + 2x + C$ ✓ Correct
D. $\dfrac{x^3}{3} - \dfrac{3x^2}{2} + 2 + C$
Solution: Term by term: x³/3 − 3x²/2 + 2x + C.
Q17 — Power Rule & Standard Integrals · easy · numerical
$\int \dfrac{3}{x}\,dx =$
A. $\dfrac{3}{x^2} + C$
B. $3x + C$
C. $\ln|3x| + C$
D. $3\ln|x| + C$ ✓ Correct
Solution: Constant factors out: 3∫(1/x)dx = 3 ln|x| + C.
Q18 — Power Rule & Standard Integrals · easy · numerical
$\int x^{-3}\,dx =$
A. $-3x^{-4} + C$
B. $\dfrac{1}{2x^2} + C$
C. $-\dfrac{1}{2x^2} + C$ ✓ Correct
D. $\dfrac{x^{-2}}{-2} \cdot (-1) + C$
Solution: ∫x⁻³dx = x⁻²/(−2) + C = −1/(2x²) + C.
Q19 — Power Rule & Standard Integrals · medium · numerical
$\int \left(4x^3 + \dfrac{2}{x}\right)dx =$
A. $4x^4 + 2\ln|x| + C$
B. $12x^2 - \dfrac{2}{x^2} + C$
C. $x^4 + 2\ln|x| + C$ ✓ Correct
D. $x^4 + 2x + C$
Solution: ∫4x³dx = x⁴; ∫(2/x)dx = 2ln|x|. Total: x⁴ + 2ln|x| + C.
Q20 — Power Rule & Standard Integrals · medium · numerical
$\int (x+1)^2\,dx$ (expand first) $=$
A. $x^3 + x^2 + x + C$
B. $2(x+1) + C$
C. $\dfrac{x^3}{3} + x^2 + x + C$ ✓ Correct
D. $\dfrac{(x+1)^3}{3} + C$ only if you skip expansion... but expanding gives the same
Solution: (x+1)² = x² + 2x + 1 ⇒ ∫ = x³/3 + x² + x + C. (The substitution form (x+1)³/3 + C is equal to this up to the constant.)
Q21 — Power Rule & Standard Integrals · medium · numerical
$\int \dfrac{x^2 + 1}{x}\,dx$ (divide first) $=$
A. $x + \dfrac{1}{x} + C$
B. $\dfrac{x^2}{2} + \ln|x| + C$ ✓ Correct
C. $\dfrac{x^2}{2} + \dfrac{1}{x} + C$
D. $\ln(x^2+1) + C$
Solution: (x²+1)/x = x + 1/x ⇒ ∫ = x²/2 + ln|x| + C. Trap: do NOT use ln of the numerator.
Q22 — Power Rule & Standard Integrals · medium · numerical
$\int 6x^{1/2}\,dx =$
A. $9x^{3/2} + C$
B. $4x^{3/2} + C$ ✓ Correct
C. $6x^{3/2} + C$
D. $3x^{3/2} + C$
Solution: 6 · x^{3/2}/(3/2) + C = 6 · (2/3)x^{3/2} + C = 4x^{3/2} + C.
Q23 — Power Rule & Standard Integrals · medium · numerical
$\int \left(\sqrt{x} + \dfrac{1}{\sqrt{x}}\right)dx =$
A. $\dfrac{2}{3}x^{3/2} + \dfrac{1}{2}x^{-1/2} + C$
B. $\dfrac{3}{2}x^{3/2} + 2\sqrt{x} + C$
C. $x^{3/2} + x^{1/2} + C$
D. $\dfrac{2}{3}x^{3/2} + 2x^{1/2} + C$ ✓ Correct
Solution: ∫x^{1/2} = (2/3)x^{3/2}; ∫x^{−1/2} = 2x^{1/2}. Sum: (2/3)x^{3/2} + 2√x + C.
Q24 — Power Rule & Standard Integrals · medium · numerical
If $\dfrac{dy}{dx} = 3x^2 - 2$ and $y = 4$ when $x = 1$, then $y =$
A. $x^3 - 2x + 4$
B. $x^3 - 2x - 5$
C. $x^3 - 2x + 5$ ✓ Correct
D. $6x + C$
Solution: y = ∫(3x²−2)dx = x³ − 2x + C. Using y(1) = 4: 1 − 2 + C = 4 ⇒ C = 5. So y = x³ − 2x + 5.
Q25 — Power Rule & Standard Integrals · medium · numerical
$\int \left(\dfrac{2}{x^3} - x\right)dx =$
A. $-\dfrac{1}{x^2} - \dfrac{x^2}{2} + C$ ✓ Correct
B. $\dfrac{1}{x^2} - \dfrac{x^2}{2} + C$
C. $-\dfrac{2}{x^2} - \dfrac{x^2}{2} + C$
D. $-\dfrac{1}{x^2} + \dfrac{x^2}{2} + C$
Solution: ∫2x⁻³dx = 2·x⁻²/(−2) = −1/x²; ∫x dx = x²/2. Result: −1/x² − x²/2 + C.