Questions: Introduction to integration
Before attempting these questions, it is highly recommended that you read Guide: Introduction to integration.
Q1
Using the power rule and laws of indices (as appropriate), find the following indefinite integrals.
1.1. \(\displaystyle \quad \int x^4 \, \mathrm{d}x\)
1.2. \(\displaystyle \quad \int 2x \, \mathrm{d}x\)
1.3. \(\displaystyle \quad \int 7x^5 \, \mathrm{d}x\)
1.4. \(\displaystyle \quad \int -5 \, \mathrm{d}t\)
1.5. \(\displaystyle \quad \int \dfrac{3}{y^3} \, \mathrm{d}y\)
1.6. \(\displaystyle \quad \int 6x^{-4} \, \mathrm{d}x\)
1.7. \(\displaystyle \quad \int -\dfrac{2}{x^5} \, \mathrm{d}x\)
1.8. \(\displaystyle \quad \int \dfrac{8}{3x^6} \, \mathrm{d}x\)
1.9. \(\displaystyle \quad \int -\dfrac{7}{2z^7} \, \mathrm{d}z\)
1.10. \(\displaystyle \quad \int x^{1/3} \, \mathrm{d}x\)
1.11. \(\displaystyle \quad \int 3t^{-2/3} \, \mathrm{d}t\)
1.12. \(\displaystyle \quad \int \dfrac{4x^{1/4}}{3} \, \mathrm{d}x\)
1.13. \(\displaystyle \quad \int \dfrac{2}{5x^{1/3}} \, \mathrm{d}x\)
1.14. \(\displaystyle \quad \int \dfrac{5}{6y^{-4/3}} \, \mathrm{d}y\)
Q2
Find the following integrals.
2.1. \(\displaystyle \quad \int e^{2x} \, \mathrm{d}x\)
2.2. \(\displaystyle \quad \int -3e^{-3x} \, \mathrm{d}x\)
2.3. \(\displaystyle \quad \int 2e^{11x} \, \mathrm{d}x\)
2.4. \(\displaystyle \quad \int \frac{4}{x} \, \mathrm{d}x\)
2.5. \(\displaystyle \quad \int -\frac{5}{3x} \, \mathrm{d}x\)
2.6. \(\displaystyle \quad \int \cos (x) \, \mathrm{d}x\)
2.7. \(\displaystyle \quad \int \sin ( 2x ) \, \mathrm{d}x\)
2.8. \(\displaystyle \quad \int \dfrac{5}{6} \cos ( x ) \, \mathrm{d}x\)
2.9. \(\displaystyle \quad \int \cos ( 3x ) \, \mathrm{d}x\)
2.10. \(\displaystyle \quad \int \sin \left( \dfrac{x}{3} \right) \, \mathrm{d}x\)
Q3
Evaluate the following definite integrals with respect to \(x\).
3.1. \(\displaystyle \quad \int_{1}^{4} 2 \, \mathrm{d}x\)
3.2. \(\displaystyle \quad \int_{-2}^{2} 3x \, \mathrm{d}x\)
3.3. \(\displaystyle \quad \int_{2}^{4} 2x^3 \, \mathrm{d}x\)
3.4. \(\displaystyle \quad \int_{1}^{27} \dfrac{4}{\sqrt[3]{x}} \, \mathrm{d}x\)
3.5. \(\displaystyle \quad \int_{0}^{\ln(3)} 4e^x \, \mathrm{d}x\)
3.6. \(\displaystyle \quad \int_{0}^{5} e^{-3x} \, \mathrm{d}x\)
3.7. \(\displaystyle \quad \int_{1}^{2} -4e^{4x} \, \mathrm{d}x\)
3.8. \(\displaystyle \quad \int_{1}^{2} \frac{2}{x} \, \mathrm{d}x\)
3.9. \(\displaystyle \quad \int_{1}^{e^3} -\frac{4}{x} \, \mathrm{d}x\)
3.10. \(\displaystyle \quad \int_{e^3}^{e^9} \frac{9}{5x} \, \mathrm{d}x\)
3.11. \(\displaystyle \quad \int_{0}^{\pi/2} \sin ( x ) \, \mathrm{d}x\)
3.12. \(\displaystyle \quad \int_{0}^{\pi} \cos ( x ) \, \mathrm{d}x\)
3.13. \(\displaystyle \quad \int_{0}^{\pi/4} \sin ( 2x ) \, \mathrm{d}x\)
3.14. \(\displaystyle \quad \int_{0}^{\pi/6} \cos(2x) \, \mathrm{d}x\)
3.15. \(\displaystyle \quad \int_{-\pi/4}^{0} \sin(3x) \, \mathrm{d}x\)
After attempting the questions above, please click this link to find the answers.
Version history and licensing
v1.0: initial version created 05/25 by Donald Campbell as part of a University of St Andrews VIP project.