Answers: Introduction to integration
These are the answers to Questions: Introduction to integration.
Please attempt the questions before reading these answers!
Q1
1.1. \(\displaystyle \quad \int x^4 \, \mathrm{d}x = \dfrac{1}{5}x^5 + C\).
1.2. \(\displaystyle \quad \int 2x \, \mathrm{d}x = x^2 + C\).
1.3. \(\displaystyle \quad \int 7x^5 \, \mathrm{d}x = \dfrac{7}{6}x^6 + C\).
1.4. \(\displaystyle \quad \int -5 \, \mathrm{d}t = -5t + C\).
1.5. \(\displaystyle \quad \int \dfrac{3}{y^3} \, \mathrm{d}y = -\dfrac{3}{2}y^{-2} + C\).
1.6. \(\displaystyle \quad \int 6x^{-4} \, \mathrm{d}x = -2x^{-3} + C\).
1.7. \(\displaystyle \quad \int -\dfrac{2}{x^5} \, \mathrm{d}x = \dfrac{1}{2}x^{-4} + C\).
1.8. \(\displaystyle \quad \int \dfrac{8}{3x^6} \, \mathrm{d}x = -\dfrac{8}{15}x^{-5} + C\).
1.9. \(\displaystyle \quad \int -\dfrac{7}{2z^7} \, \mathrm{d}z = \dfrac{7}{12}z^{-6} + C\).
1.10. \(\displaystyle \quad \int x^{1/3} \, \mathrm{d}x = \dfrac{3}{4} x^{4/3} + C\).
1.11. \(\displaystyle \quad \int 3t^{-2/3} \, \mathrm{d}t = 9t^{1/3} + C\).
1.12. \(\displaystyle \quad \int \dfrac{4x^{1/4}}{3} \, \mathrm{d}x = \dfrac{16}{15} x^{5/4} + C\).
1.13. \(\displaystyle \quad \int \dfrac{2}{5x^{1/3}} \, \mathrm{d}x = \dfrac{3}{5} x^{2/3} + C\).
1.14. \(\displaystyle \quad \int \dfrac{5}{6y^{-4/3}} \, \mathrm{d}y = \dfrac{5}{14} y^{7/3} + C\).
Q5
2.1. \(\displaystyle \quad \int e^{2x} \, \mathrm{d}x = \frac{1}{2}e^{2x} + C\)
2.2. \(\displaystyle \quad \int -3e^{-3x} \, \mathrm{d}x = e^{-3x} + C\)
2.3. \(\displaystyle \quad \int 2e^{11x} \, \mathrm{d}x = \frac{2}{11}e^{11x} + C\)
2.4. \(\displaystyle \quad \int \frac{4}{x} \, \mathrm{d}x = 4\ln|x| + C\)
2.5. \(\displaystyle \quad \int -\frac{5}{3x} \, \mathrm{d}x = -\frac{5}{3}\ln|x| + C\)
2.6. \(\displaystyle \quad \int \cos (x) \, \mathrm{d}x = \sin ( x ) + C\).
2.7. \(\displaystyle \quad \int \sin ( 2x ) \, \mathrm{d}x = -\dfrac{1}{2} \cos ( 2x ) + C\).
2.8. \(\displaystyle \quad \int \dfrac{5}{6} \cos ( x ) \, \mathrm{d}x = \dfrac{5}{6} \sin ( x ) + C\).
2.9. \(\displaystyle \quad \int \cos ( 3x ) \, \mathrm{d}x = \dfrac{1}{3} \sin ( 3x ) + C\).
2.10. \(\displaystyle \quad \int \sin \left( \dfrac{x}{3} \right) \, \mathrm{d}x = -3 \cos \left( \dfrac{x}{3} \right) + C\).
Q3
3.1. \(\displaystyle \quad \int_{1}^{4} 2 \, \mathrm{d}x = 6\)
3.2. \(\displaystyle \quad \int_{-2}^{2} 3x \, \mathrm{d}x = 0\)
3.3. \(\displaystyle \quad \int_{2}^{4} 2x^3 \, \mathrm{d}x = 120\)
3.4. \(\displaystyle \quad \int_{1}^{27} \dfrac{4}{\sqrt[3]{x}} \, \mathrm{d}x = 48\)
3.5. \(\displaystyle \quad \int_{0}^{\ln(3)} 4e^x \, \mathrm{d}x = 8\)
3.6. \(\displaystyle \quad \int_{0}^{5} e^{-3x} \, \mathrm{d}x = \frac{1}{3}\left(1 - e^{-15}\right)\)
3.7. \(\displaystyle \quad \int_{1}^{2} -4e^{4x} \, \mathrm{d}x = e^4(1-e^4)\)
3.8. \(\displaystyle \quad \int_{1}^{2} \frac{2}{x} \, \mathrm{d}x = 2\ln(2)\)
3.9. \(\displaystyle \quad \int_{1}^{e^3} -\frac{4}{x} \, \mathrm{d}x = -12\)
3.10. \(\displaystyle \quad \int_{e^3}^{e^9} \frac{9}{5x} \, \mathrm{d}x = \frac{54}{5}\)
3.11. \(\displaystyle \quad \int_{0}^{\pi/2} \sin ( x ) \, \mathrm{d}x = 1\)
3.12. \(\displaystyle \quad \int_{0}^{\pi} \cos ( x ) \, \mathrm{d}x = 0\)
3.13. \(\displaystyle \quad \int_{0}^{\pi/4} \sin ( 2x ) \, \mathrm{d}x = \dfrac{1}{2}\)
3.14. \(\displaystyle \quad \int_{0}^{\pi/6} \cos(2x) \, \mathrm{d}x = \dfrac{\sqrt{3}}{4}\)
3.15. \(\displaystyle \quad \int_{-\pi/4}^{0} \sin(3x) \, \mathrm{d}x = -\dfrac{1}{3} - \dfrac{1}{3\sqrt{2}}\)
Version history and licensing
v1.0: initial version created 05/25 by Donald Campbell as part of a University of St Andrews VIP project.