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Exponential & Trigonometric Functions — JEE Main Basic Maths MCQs with Solutions

Free JEE Main Basic Maths Exponential & Trigonometric Functions 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 — Exponential & Trigonometric Functions · easy · theory
$\int e^x\,dx =$
A. $e^x + C$  ✓ Correct
B. $\dfrac{e^{x+1}}{x+1} + C$
C. $\dfrac{e^x}{x} + C$
D. $x e^{x-1} + C$
Solution: The exponential is its own integral: ∫eˣdx = eˣ + C.
Q2 — Exponential & Trigonometric Functions · easy · theory
$\int e^{ax}\,dx =$
A. $\dfrac{e^{ax}}{a^2} + C$
B. $\dfrac{e^{ax}}{a} + C$  ✓ Correct
C. $a e^{ax} + C$
D. $e^{ax} + C$
Solution: Divide by the coefficient of x: ∫e^{ax}dx = e^{ax}/a + C.
Q3 — Exponential & Trigonometric Functions · easy · theory
$\int \sin x\,dx =$
A. $-\cos x + C$  ✓ Correct
B. $\sin x + C$
C. $-\sin x + C$
D. $\cos x + C$
Solution: ∫sin x dx = −cos x + C. Sign trap: integrating sine introduces a MINUS sign.
Q4 — Exponential & Trigonometric Functions · easy · theory
$\int \cos x\,dx =$
A. $-\sin x + C$
B. $\cos x + C$
C. $-\cos x + C$
D. $\sin x + C$  ✓ Correct
Solution: ∫cos x dx = sin x + C (no sign change for cosine).
Q5 — Exponential & Trigonometric Functions · easy · theory
$\int \sec^2 x\,dx =$
A. $\sec x + C$
B. $\sec x\tan x + C$
C. $-\cot x + C$
D. $\tan x + C$  ✓ Correct
Solution: Since d/dx(tan x) = sec²x, ∫sec²x dx = tan x + C.
Q6 — Exponential & Trigonometric Functions · easy · theory
$\int \csc^2 x\,dx =$
A. $-\cot x + C$  ✓ Correct
B. $\cot x + C$
C. $-\csc x + C$
D. $\tan x + C$
Solution: Since d/dx(cot x) = −csc²x, ∫csc²x dx = −cot x + C.
Q7 — Exponential & Trigonometric Functions · medium · theory
To integrate sin²x, the correct first step is to use the identity:
A. $\sin^2 x = \dfrac{1 + \cos 2x}{2}$
B. $\sin^2 x = 2\sin x\cos x$
C. $\sin^2 x = \dfrac{1 - \cos 2x}{2}$  ✓ Correct
D. $\sin^2 x = 1 + \cos 2x$
Solution: sin²x = (1 − cos2x)/2 turns the square into terms that integrate directly. (cos²x uses the + version.)
Q8 — Exponential & Trigonometric Functions · easy · theory
$\int \cos(ax)\,dx =$
A. $a\sin(ax) + C$
B. $-\dfrac{\sin(ax)}{a} + C$
C. $\dfrac{\sin(ax)}{a} + C$  ✓ Correct
D. $\sin(ax) + C$
Solution: ∫cos(ax)dx = sin(ax)/a + C — divide by the inner coefficient a.
Q9 — 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. $-\sin x + C$
C. $-\cos x + C$ (a minus sign is required)  ✓ Correct
D. $\cos x + C$
Solution: Because d/dx(−cos x) = sin x, the integral of sine is −cos x + C — the sign is the classic trap.
Q10 — Exponential & Trigonometric Functions · medium · theory
For radioactive-type decay, $\int e^{-\lambda t}\,dt$ equals:
A. $\dfrac{1}{\lambda}e^{-\lambda t} + C$
B. $-\lambda e^{-\lambda t} + C$
C. $e^{-\lambda t} + C$
D. $-\dfrac{1}{\lambda}e^{-\lambda t} + C$  ✓ Correct
Solution: ∫e^{−λt}dt = e^{−λt}/(−λ) + C = −(1/λ)e^{−λt} + C. The negative inner coefficient brings a minus sign.
Q11 — Exponential & Trigonometric Functions · easy · numerical
$\int e^{2x}\,dx =$
A. $\dfrac{e^{2x}}{2} + C$  ✓ Correct
B. $e^{2x} + C$
C. $\dfrac{e^{2x}}{4} + C$
D. $2e^{2x} + C$
Solution: ∫e^{2x}dx = e^{2x}/2 + C.
Q12 — Exponential & Trigonometric Functions · easy · numerical
$\int \sin 3x\,dx =$
A. $-\cos 3x + C$
B. $-3\cos 3x + C$
C. $-\dfrac{\cos 3x}{3} + C$  ✓ Correct
D. $\dfrac{\cos 3x}{3} + C$
Solution: ∫sin(3x)dx = −cos(3x)/3 + C (sign from sine, divide by 3).
Q13 — Exponential & Trigonometric Functions · easy · numerical
$\int \cos 5x\,dx =$
A. $\sin 5x + C$
B. $5\sin 5x + C$
C. $\dfrac{\sin 5x}{5} + C$  ✓ Correct
D. $-\dfrac{\sin 5x}{5} + C$
Solution: ∫cos(5x)dx = sin(5x)/5 + C.
Q14 — Exponential & Trigonometric Functions · easy · numerical
$\int \sec^2(2x)\,dx =$
A. $-\dfrac{\cot 2x}{2} + C$
B. $\dfrac{\tan 2x}{2} + C$  ✓ Correct
C. $2\tan 2x + C$
D. $\tan 2x + C$
Solution: ∫sec²(2x)dx = tan(2x)/2 + C.
Q15 — Exponential & Trigonometric Functions · easy · numerical
$\int e^{-x}\,dx =$
A. $-e^{-x} + C$  ✓ Correct
B. $e^{x} + C$
C. $-\dfrac{e^{-x}}{2} + C$
D. $e^{-x} + C$
Solution: ∫e^{−x}dx = e^{−x}/(−1) + C = −e^{−x} + C.
Q16 — Exponential & Trigonometric Functions · easy · numerical
$\int (\sin x + \cos x)\,dx =$
A. $\sin x + \cos x + C$
B. $\cos x - \sin x + C$
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.
Q17 — Exponential & Trigonometric Functions · medium · numerical
$\int \sin^2 x\,dx =$
A. $-\dfrac{\cos^3 x}{3} + C$
B. $\dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$  ✓ Correct
C. $\dfrac{\sin^3 x}{3} + C$
D. $\dfrac{x}{2} + \dfrac{\sin 2x}{4} + C$
Solution: sin²x = (1−cos2x)/2 ⇒ ∫ = x/2 − (sin2x)/4 + C.
Q18 — Exponential & Trigonometric Functions · medium · numerical
$\int \cos^2 x\,dx =$
A. $\dfrac{x}{2} - \dfrac{\sin 2x}{4} + C$
B. $\dfrac{x}{2} + \dfrac{\sin 2x}{4} + C$  ✓ Correct
C. $\dfrac{\sin^3 x}{3} + C$
D. $\dfrac{\cos^3 x}{3} + C$
Solution: cos²x = (1+cos2x)/2 ⇒ ∫ = x/2 + (sin2x)/4 + C.
Q19 — Exponential & Trigonometric Functions · easy · numerical
$\int 3e^{2x}\,dx =$
A. $\dfrac{3}{2}e^{2x} + C$  ✓ Correct
B. $3e^{2x} + C$
C. $\dfrac{3}{4}e^{2x} + C$
D. $6e^{2x} + C$
Solution: 3 · e^{2x}/2 + C = (3/2)e^{2x} + C.
Q20 — Exponential & Trigonometric Functions · medium · numerical
$\int \csc^2(3x)\,dx =$
A. $-\dfrac{\cot 3x}{3} + C$  ✓ Correct
B. $\dfrac{\cot 3x}{3} + C$
C. $-3\cot 3x + C$
D. $\dfrac{\tan 3x}{3} + C$
Solution: ∫csc²(3x)dx = −cot(3x)/3 + C.
Q21 — Exponential & Trigonometric Functions · medium · numerical
$\int (2\cos x - 3\sin x)\,dx =$
A. $-2\sin x + 3\cos x + C$
B. $2\cos x + 3\sin x + C$
C. $2\sin x - 3\cos x + C$
D. $2\sin x + 3\cos x + C$  ✓ Correct
Solution: ∫2cos x = 2sin x; ∫(−3sin x) = +3cos x. Result: 2sin x + 3cos x + C.
Q22 — Exponential & Trigonometric Functions · easy · numerical
$\int e^{5x}\,dx =$
A. $e^{5x} + C$
B. $\dfrac{e^{5x}}{25} + C$
C. $5e^{5x} + C$
D. $\dfrac{e^{5x}}{5} + C$  ✓ Correct
Solution: ∫e^{5x}dx = e^{5x}/5 + C.
Q23 — Exponential & Trigonometric Functions · medium · numerical
$\int \sin\dfrac{x}{2}\,dx =$
A. $-\cos\dfrac{x}{2} + C$
B. $-2\cos\dfrac{x}{2} + C$  ✓ Correct
C. $-\dfrac{1}{2}\cos\dfrac{x}{2} + C$
D. $2\cos\dfrac{x}{2} + C$
Solution: a = 1/2 ⇒ ∫sin(x/2)dx = −cos(x/2)/(1/2) + C = −2cos(x/2) + C.
Q24 — Exponential & Trigonometric Functions · easy · numerical
$\int \left(e^x + \dfrac{1}{x}\right)dx =$
A. $e^x - \dfrac{1}{x^2} + C$
B. $xe^{x-1} + \ln|x| + C$
C. $e^x + \ln x^2 + C$
D. $e^x + \ln|x| + C$  ✓ Correct
Solution: ∫eˣdx = eˣ; ∫(1/x)dx = ln|x|. Sum: eˣ + ln|x| + C.
Q25 — Exponential & Trigonometric Functions · medium · numerical
$\int \sec x\tan x\,dx =$
A. $\sec x + C$  ✓ Correct
B. $\sec^2 x + C$
C. $\tan x + C$
D. $-\sec x + C$
Solution: Since d/dx(sec x) = sec x tan x, the integral is sec x + C.