Elementary Mathematics II — 2025B Past Questions

University of Ilorin MTH102: 20 questions.

Topics: Curve sketching, integration, definite integrals, reduction formulae., Derivatives as limits of rate of change, differentiation techniques, maxima and minima., Applications to areas, volumes (Trapezium and Simpson's rules).

1. Evaluate the definite integral $\int_{0}^{-2} (x^3 + 3x^2 - 1) dx$.
   1. $2$
   2. $10$
   3. $-2$
   4. $6$
2. Evaluate the indefinite integral $\int \sin(-6x) dx$.
   1. $-\frac{1}{6} \cos(-6x) + C$
   2. $\frac{1}{6} \cos(-6x) + C$
   3. $6 \cos(-6x) + C$
   4. $-6 \cos(-6x) + C$
3. Evaluate the integral $\int \left(x^3 + \frac{1}{\sqrt{4-x^2}}\right) dx$.
   1. $\frac{x^4}{4} + \sin^{-1}\left(\frac{x}{2}\right) + C$
   2. $\frac{x^4}{4} + \tan^{-1}\left(\frac{x}{2}\right) + C$
   3. $x^4 + \sin^{-1}\left(\frac{x}{2}\right) + C$
   4. $x^4 + \tan^{-1}\left(\frac{x}{2}\right) + C$
4. Evaluate the integral $\int (5 - 4x^3) dx$.
   1. $5x - x^4 + C$
   2. $5x - 4x^4 + C$
   3. $5x - 12x^4 + C$
   4. $5 - 12x^2 + C$
5. Evaluate the definite integral $\int_{0}^{1} \frac{1}{\sqrt{4-x^2}} dx$.
   1. $60^\circ$
   2. $90^\circ$
   3. $30^\circ$
   4. $0^\circ$
6. Find the derivative of $y = \log_e(1/x)$.
   1. $-x$
   2. $1/x$
   3. $-1/x$
   4. $x$
7. Evaluate the definite integral $\int_{-\pi/2}^{\pi/2} \cos x dx$.
   1. $1$
   2. $-1$
   3. $2$
   4. $-2$
8. Evaluate the definite integral $\int_{-1}^{1} x^2(x - x^2) dx$.
   1. $1/5$
   2. $2/5$
   3. $-2/5$
   4. $-1/5$
9. Find the second derivative of $y = e^{\sin t}$.
   1. $e^{\sin t}(\cos^2 t - \sin t)$
   2. $e^{\sin t}(-\cos^2 t - \sin t)$
   3. $e^{\sin t}(\cos^2 t - \sin^2 t)$
   4. $e^{\sin t}(\cos t - \sin t)$
10. Evaluate the integral $\int 2\cos^2 \theta d\theta$.
    1. $\frac{\sin 2\theta}{2} + \theta + C$
    2. $\frac{\cos 2\theta}{2} + \theta + C$
    3. $\sin 2\theta + \theta + C$
    4. $\cos 2\theta - \theta + C$
11. Evaluate the integral $\int 2\sin^2 \theta d\theta$.
    1. $\theta - \frac{\sin 2\theta}{2} + C$
    2. $\theta + \frac{\sin 2\theta}{2} + C$
    3. $\frac{\cos 2\theta}{2} + \theta + C$
    4. $\theta - \cos 2\theta + C$
12. Find the area under the curve bounded by the intersection of $y = x^2$ and $y = x$.
    1. $1/6$
    2. $-1/6$
    3. $1/3$
    4. $-1/3$
13. Evaluate the integral $\int \cos 3x \sin 2x dx$.
    1. $\frac{\cos x}{2} - \frac{\cos 5x}{10} + C$
    2. $\frac{\sin x}{2} - \frac{\sin 5x}{10} + C$
    3. $-\frac{\cos x}{2} + \frac{\cos 5x}{10} + C$
    4. $\frac{\cos x}{2} + \frac{\cos 5x}{10} + C$
14. Evaluate the integral $\int \left(5x^{3/2} - 2\csc^2 x\right) dx$.
    1. $2x^{5/2} + 2\cot x + C$
    2. $2x^{5/2} - 2\cot x + C$
    3. $\frac{5}{2}x^{5/2} + 2\cot x + C$
    4. $2x^{5/2} + 2\tan x + C$
15. Evaluate the integral $\int \frac{3x^3 + 4x^2 - 7x}{2} dx$.
    1. $\frac{3x^4}{8} + \frac{2x^3}{3} - \frac{7x^2}{4} + C$
    2. $\frac{3x^4}{4} + \frac{4x^3}{3} - \frac{7x^2}{2} + C$
    3. $\frac{3x^4}{8} + \frac{2x^3}{3} + \frac{7x^2}{4} + C$
    4. $\frac{3x^4}{2} + \frac{4x^3}{3} - \frac{7x^2}{4} + C$
16. Evaluate the definite integral $\int_{1}^{2} 2x(x^2 + 1) dx$.
    1. $21/2$
    2. $15/2$
    3. $11/2$
    4. $9/2$
17. Evaluate the integral $\int (x^2 - 1)^4 dx$.
    1. $\frac{x^9}{9} - \frac{4x^7}{7} + \frac{6x^5}{5} - \frac{4x^3}{3} + x + C$
    2. $\frac{x^9}{9} + \frac{4x^7}{7} + \frac{6x^5}{5} + \frac{4x^3}{3} + x + C$
    3. $\frac{(x^2 - 1)^5}{10x} + C$
    4. $\frac{x^9}{9} - \frac{4x^7}{7} + \frac{6x^5}{5} - x + C$
18. Evaluate the integral $\int \frac{dx}{(5+2x)^3}$.
    1. $\frac{-1}{4(5+2x)^2} + C$
    2. $\frac{-1}{2(5+2x)^2} + C$
    3. $\frac{1}{4(5+2x)^2} + C$
    4. $\frac{-1}{8(5+2x)^2} + C$
19. Evaluate the integral $\int \frac{x^2+1}{x^3+3x+1} dx$.
    1. $\frac{1}{3}\ln|x^3+3x+1| + C$
    2. $3\ln|x^3+3x+1| + C$
    3. $\ln|x^3+3x+1| + C$
    4. $\frac{1}{2}\ln|x^2+1| + C$
20. Evaluate the integral $\int x^2 e^x dx$.
    1. $e^x(x^2 - 2x + 2) + C$
    2. $e^x(x^2 + 2x + 2) + C$
    3. $e^x(x^2 - 2x - 2) + C$
    4. $x^2 e^x - 2e^x + C$

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