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Integration — MH-CET Basic Maths MCQs with Solutions
Free MH-CET Basic Maths Integration MCQs with step-by-step solutions covering Power Rule & Standard Integrals, Exponential & Trigonometric Functions, Definite Integrals & Area Under Curve, Substitution & Physics Applications. Practise online on Prepizo — no login needed.
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Sample questions with solutions
Q1 — Power Rule & Standard Integrals · easy · theory
The power rule for integration states $\int x^n\,dx =$
A. $x^{n+1} + C$
B. $\dfrac{x^{n-1}}{n-1} + C$
C. $n x^{n-1} + C$
D. $\dfrac{x^{n+1}}{n+1} + C$ (for $n \neq -1$) ✓ Correct
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. $\dfrac{x^0}{0} + C$
C. $\ln|x| + C$ ✓ Correct
D. $-x^{-2} + C$
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. Any constant differentiates to zero, so antiderivatives are unique only up to a constant ✓ Correct
B. It makes the integral larger
C. It is required only for definite integrals
D. It represents the area
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. $\int f(x)\,dx + \int g(x)\,dx$ only
B. $\int f(x)\,dx \cdot \int g(x)\,dx$
C. $a\int f(x)\,dx + b\int g(x)\,dx$ ✓ Correct
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. Same as differentiation
C. Reverse process of differentiation (antiderivative) ✓ Correct
D. Process of finding a slope
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. $0$
D. $k + C$
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. $\ln x$
B. $x^2$
C. $x^{-1/2}$
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 · easy · theory
The integral $\int (3x^2 - 4x + 5)\,dx$ is found by:
A. Differentiating each term
B. Integrating each term separately using linearity ✓ Correct
C. Applying the quotient rule
D. Multiplying the terms first
Solution: By linearity, integrate term by term: x³ − 2x² + 5x + C.
Q9 — Power Rule & Standard Integrals · easy · numerical
$\int x^3\,dx =$
A. $3x^2 + C$
B. $4x^4 + C$
C. $\dfrac{x^4}{3} + C$
D. $\dfrac{x^4}{4} + C$ ✓ Correct
Solution: ∫x³dx = x⁴/4 + C.
Q10 — Power Rule & Standard Integrals · easy · numerical
$\int x^5\,dx =$
A. $\dfrac{x^6}{6} + C$ ✓ Correct
B. $5x^4 + C$
C. $\dfrac{x^6}{5} + C$
D. $\dfrac{x^4}{4} + C$
Solution: ∫x⁵dx = x⁶/6 + C.
Q11 — Power Rule & Standard Integrals · easy · numerical
$\int (2x + 3)\,dx =$
A. $x^2 + 3 + C$
B. $x^2 + 3x + C$ ✓ Correct
C. $2 + C$
D. $2x^2 + 3x + C$
Solution: ∫2x dx + ∫3 dx = x² + 3x + C.
Q12 — Power Rule & Standard Integrals · easy · numerical
$\int \dfrac{1}{x^2}\,dx =$
A. $-\dfrac{1}{x} + C$ ✓ Correct
B. $-\dfrac{2}{x^3} + C$
C. $\dfrac{1}{x} + C$
D. $\ln x^2 + C$
Solution: ∫x⁻²dx = x⁻¹/(−1) + C = −1/x + C.
Q13 — Power Rule & Standard Integrals · easy · numerical
$\int \sqrt{x}\,dx =$
A. $2x^{1/2} + C$
B. $\dfrac{3}{2}x^{3/2} + C$
C. $\dfrac{1}{2}x^{-1/2} + C$
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.
Q14 — Power Rule & Standard Integrals · easy · numerical
$\int (x^2 - 3x + 2)\,dx =$
A. $\dfrac{x^3}{3} - 3x^2 + 2x + C$
B. $\dfrac{x^3}{3} - \dfrac{3x^2}{2} + 2x + C$ ✓ Correct
C. $2x - 3 + C$
D. $\dfrac{x^3}{3} - \dfrac{3x^2}{2} + 2 + C$
Solution: Term by term: x³/3 − 3x²/2 + 2x + C.
Q15 — Power Rule & Standard Integrals · easy · numerical
$\int \dfrac{3}{x}\,dx =$
A. $3x + C$
B. $\ln|3x| + C$
C. $\dfrac{3}{x^2} + C$
D. $3\ln|x| + C$ ✓ Correct
Solution: Constant factors out: 3∫(1/x)dx = 3 ln|x| + C.
Q16 — Power Rule & Standard Integrals · easy · numerical
$\int x^{-3}\,dx =$
A. $-\dfrac{1}{2x^2} + C$ ✓ Correct
B. $\dfrac{1}{2x^2} + C$
C. $-3x^{-4} + C$
D. $\dfrac{x^{-2}}{-2} \cdot (-1) + C$
Solution: ∫x⁻³dx = x⁻²/(−2) + C = −1/(2x²) + C.
Q17 — Exponential & Trigonometric Functions · easy · theory
$\int e^x\,dx =$
A. $\dfrac{e^x}{x} + C$
B. $x e^{x-1} + C$
C. $\dfrac{e^{x+1}}{x+1} + C$
D. $e^x + C$ ✓ Correct
Solution: The exponential is its own integral: ∫eˣdx = eˣ + C.
Q18 — Exponential & Trigonometric Functions · easy · theory
$\int e^{ax}\,dx =$
A. $a e^{ax} + C$
B. $\dfrac{e^{ax}}{a} + C$ ✓ Correct
C. $e^{ax} + C$
D. $\dfrac{e^{ax}}{a^2} + C$
Solution: Divide by the coefficient of x: ∫e^{ax}dx = e^{ax}/a + C.
Q19 — Exponential & Trigonometric Functions · easy · theory
$\int \sin x\,dx =$
A. $-\sin x + C$
B. $\sin x + C$
C. $-\cos x + C$ ✓ Correct
D. $\cos x + C$
Solution: ∫sin x dx = −cos x + C. Sign trap: integrating sine introduces a MINUS sign.
Q20 — Exponential & Trigonometric Functions · easy · theory
$\int \cos x\,dx =$
A. $-\cos x + C$
B. $\cos x + C$
C. $-\sin x + C$
D. $\sin x + C$ ✓ Correct
Solution: ∫cos x dx = sin x + C (no sign change for cosine).
Q21 — Exponential & Trigonometric Functions · easy · theory
$\int \sec^2 x\,dx =$
A. $\sec x\tan x + C$
B. $\tan x + C$ ✓ Correct
C. $-\cot x + C$
D. $\sec x + C$
Solution: Since d/dx(tan x) = sec²x, ∫sec²x dx = tan x + C.
Q22 — Exponential & Trigonometric Functions · easy · theory
$\int \csc^2 x\,dx =$
A. $-\csc x + C$
B. $\tan x + C$
C. $-\cot x + C$ ✓ Correct
D. $\cot x + C$
Solution: Since d/dx(cot x) = −csc²x, ∫csc²x dx = −cot x + C.
Q23 — Exponential & Trigonometric Functions · easy · theory
$\int \cos(ax)\,dx =$
A. $a\sin(ax) + C$
B. $\sin(ax) + C$
C. $\dfrac{\sin(ax)}{a} + C$ ✓ Correct
D. $-\dfrac{\sin(ax)}{a} + C$
Solution: ∫cos(ax)dx = sin(ax)/a + C — divide by the inner coefficient a.
Q24 — Exponential & Trigonometric Functions · easy · theory
A common student error is to write $\int \sin x\,dx = \cos x + C$. The correct result is:
A. $\sin x + C$
B. $\cos x + C$
C. $-\cos x + C$ (a minus sign is required) ✓ Correct
D. $-\sin x + C$
Solution: Because d/dx(−cos x) = sin x, the integral of sine is −cos x + C — the sign is the classic trap.
Q25 — Exponential & Trigonometric Functions · easy · numerical
$\int e^{2x}\,dx =$
A. $\dfrac{e^{2x}}{4} + C$
B. $e^{2x} + C$
C. $2e^{2x} + C$
D. $\dfrac{e^{2x}}{2} + C$ ✓ Correct
Solution: ∫e^{2x}dx = e^{2x}/2 + C.
Q26 — Exponential & Trigonometric Functions · easy · numerical
$\int \sin 3x\,dx =$
A. $-3\cos 3x + C$
B. $-\cos 3x + C$
C. $\dfrac{\cos 3x}{3} + C$
D. $-\dfrac{\cos 3x}{3} + C$ ✓ Correct
Solution: ∫sin(3x)dx = −cos(3x)/3 + C (sign from sine, divide by 3).
Q27 — Exponential & Trigonometric Functions · easy · numerical
$\int \cos 5x\,dx =$
A. $-\dfrac{\sin 5x}{5} + C$
B. $5\sin 5x + C$
C. $\sin 5x + C$
D. $\dfrac{\sin 5x}{5} + C$ ✓ Correct
Solution: ∫cos(5x)dx = sin(5x)/5 + C.
Q28 — Exponential & Trigonometric Functions · easy · numerical
$\int \sec^2(2x)\,dx =$
A. $\tan 2x + C$
B. $\dfrac{\tan 2x}{2} + C$ ✓ Correct
C. $-\dfrac{\cot 2x}{2} + C$
D. $2\tan 2x + C$
Solution: ∫sec²(2x)dx = tan(2x)/2 + C.
Q29 — Exponential & Trigonometric Functions · easy · numerical
$\int e^{-x}\,dx =$
A. $e^{-x} + C$
B. $-\dfrac{e^{-x}}{2} + C$
C. $-e^{-x} + C$ ✓ Correct
D. $e^{x} + C$
Solution: ∫e^{−x}dx = e^{−x}/(−1) + C = −e^{−x} + C.
Q30 — Exponential & Trigonometric Functions · easy · numerical
$\int (\sin x + \cos x)\,dx =$
A. $\cos x - \sin x + C$
B. $\sin x + \cos x + C$
C. $\sin x - \cos x + C$ ✓ Correct
D. $-\sin x + \cos x + C$
Solution: ∫sin x = −cos x; ∫cos x = sin x. Sum: sin x − cos x + C.