3.0 Objectives

This chapter covers the following topics related to integral calculus. After successful completion of this section, you will be able to:

• Evaluate indefinite and definite integrals.

• Use the substitution method and integration-by-parts to evaluate integrals.

• Integration of trigonometric and hyperbolic functions.

• Derive reduction formulae and use these to evaluate integrals.

• Integrate using other methods, such as the method of partial fractions and completing the square.

• Evaluate the area under the curve and between the curves.

• Find the volume of a solid of revolution.

• Determine the lengths of plane curves.

• Find the centre of mass.

• Determine the MacLaurin series expansion for some common functions.

• Define power series and use power series to evaluate integrals.

3.1 Indefinite Integrals

• i)

  The video in the link below demonstrates how to apply the indefinite integral for the following trigonometric functions.

  \[ \begin{array}{lll} \text{a)}\quad \displaystyle \int \cos x\, dx & \quad \text{b)}\quad \displaystyle \int \sin x\, dx & \quad \text{c)}\quad \displaystyle \int \sec ^2 x\, dx \\ \text{d)}\quad \displaystyle \int \dfrac {3}{7} \sin x\, dx & \quad \text{e)}\quad \displaystyle \int \dfrac {4 \sec ^2 x}{5} \, dx & \quad \text{f)}\quad \displaystyle \int \dfrac {5}{\sec x}\, dx \\ \text{g)}\quad \displaystyle \int \dfrac {2}{7} \sin x\, dx & \quad \text{h)}\quad \displaystyle \int \dfrac {3}{8 \csc x}\, dx & \quad \text{i)}\quad \displaystyle \int \dfrac {-4 \cos x}{7} \, dx \\ \text{j)}\quad \displaystyle \int 3x - \dfrac {\sec ^2 x}{8} \, dx & \quad \text{k)}\quad \displaystyle \int 1 + \tan ^2 x\, dx & \end{array} \]

• ii)

  The video in the link below demonstrates how to apply the indefinite integral for the following exponential functions.

  \[ \begin{array}{lll} \text{a)}\quad \displaystyle \int \dfrac {2}{7} e^x \, dx & \hspace{1.45 cm} \text{b)}\quad \displaystyle \int \dfrac {3}{5} e^{2x} \, dx & \hspace{1.0 cm} \text{c)}\quad \displaystyle \int \dfrac {4}{3e^{5x}} \, dx \\ \text{d)}\quad \displaystyle \int \dfrac {2}{3} e^{3x - 2} \, dx & \hspace{1.4cm} \text{e)}\quad \displaystyle \int 3e^{7x} - \dfrac {2}{3} e^{5x - 1} \, dx & \hspace{1cm} \text{f)}\quad \displaystyle \int 3e^{5 - 2x} - \dfrac {3}{2e^{4x}} \, dx \end{array} \]

3.2 Definite Integration

• i)

  The video in the link below explains what a definite integral is and demonstrates how to apply it to the following functions.

  \[ \text{a)}\quad \int _{1}^{3} 3x^2 \, dx \qquad \qquad \text{b)}\quad \int _{-1}^{2} (4x^3 - 3)\, dx \]

• ii)

  The video below further demonstrates how to find the definite integral for algebraic expressions with various indices, including trigonometric functions.

  \[ \begin{array}{lll} \text{a)}\quad \displaystyle \int _2^4 x^3\, dx & \quad \text{b)}\quad \displaystyle \int _4^{10} 7\, dx & \quad \text{c)}\quad \displaystyle \int _1^2 (3x^2 - 5x + 2)\, dx \\[1.5em] \text{d)}\quad \displaystyle \int _{-1}^3 (2x + 3)^2\, dx & \quad \text{e)}\quad \displaystyle \int _{\frac{1}{2}}^1 \frac{1}{x^2}\, dx & \quad \text{f)}\quad \displaystyle \int _1^e \frac{1}{x}\, dx \\[1.5em] \text{g)}\quad \displaystyle \int _4^9 \sqrt{x}\, dx & \quad \text{h)}\quad \displaystyle \int _2^3 \frac{x^3 - 5x^2}{x}\, dx & \quad \text{i)}\quad \displaystyle \int _0^{\frac{1}{3}} e^{3x}\, dx \\[1.5em] \text{j)}\quad \displaystyle \int _{\frac{\pi }{6}}^{\frac{\pi }{3}} \cos x\, dx & \quad \text{k)}\quad \displaystyle \int _0^{\frac{\pi }{4}} \sin (2x)\, dx & \quad \text{l)}\quad \displaystyle \int _{\frac{\pi }{4}}^{\frac{3\pi }{4}} \sec ^2 x\, dx \\[1.5em] \text{m)}\quad \displaystyle \int _0^1 x^2(x^3 + 5)^2\, dx & \quad \text{n)}\quad \displaystyle \int _1^2 4xe^{x^2}\, dx & \quad \text{o)}\quad \displaystyle \int _0^1 xe^x\, dx \end{array} \]

3.3 Integration techniques

The following section provides worked examples of integration by substitution and integration by parts techniques for both definite and indefinite integrals.

• i)

  Integration by substitution: Indefinite integrals

  \[ \begin{array}{lll} \text{a)} \quad \displaystyle \int 4x(x^2 + 5)^3\, dx & \quad \text{b)} \quad \displaystyle \int 8\cos (4x)\, dx & \quad \text{c)} \quad \displaystyle \int x^3 e^{x^4}\, dx \\[1em] \text{d)} \quad \displaystyle \int 8\sqrt{40 - 2x^2}\, dx & \quad \text{e)} \quad \displaystyle \int \frac{x^3}{(2 + x^4)^2}\, dx & \quad \text{f)} \quad \displaystyle \int \sin ^4 x \cos x\, dx \\[1em] \text{g)} \quad \displaystyle \int \sqrt{5x + 4}\, dx & \quad \text{h)} \quad \displaystyle \int x\sqrt{3x + 2}\, dx & \quad \text{i)} \quad \displaystyle \int 2x\sqrt{4x - 5}\, dx \end{array} \]

• ii)

  Integration by substitution: Definite integrals

  \[ \begin{array}{lll} \text{a)} \quad \displaystyle \int _0^2 2x(x^2 + 4)^2\, dx & \quad \text{b)} \quad \displaystyle \int _0^4 4x\sqrt{16 - x^2}\, dx & \quad \text{c)} \quad \displaystyle \int _1^2 \frac{2x}{(1 + x^2)^3}\, dx \end{array} \]

• iii)

  Integration by parts: Indefinite integrals

  \[ \begin{array}{lll} \text{a)}\quad \displaystyle \int xe^x\, dx & \quad \text{b)}\quad \displaystyle \int x \sin x\, dx & \quad \text{c)}\quad \displaystyle \int x^2 \ln x\, dx \\[1em] \text{d)}\quad \displaystyle \int \ln x\, dx & \quad \text{e)}\quad \displaystyle \int x^2 \sin x\, dx & \quad \text{f)}\quad \displaystyle \int x \cos x\, dx \\[1em] \text{g)}\quad \displaystyle \int x^2 e^x\, dx & \quad \text{h)}\quad \displaystyle \int (\ln x)^2\, dx & \quad \text{i)}\quad \displaystyle \int \ln x^7\, dx \\[1em] \text{j)}\quad \displaystyle \int e^x \sin x\, dx & \quad \text{k)}\quad \displaystyle \int \frac{(\ln x)^2}{x}\, dx & \quad \text{l)}\quad \displaystyle \int e^{3x} \cos (4x)\, dx \end{array} \]

• iv)

  Integration by parts: Definite integrals

  \[ \begin{array}{ll} \text{a)} \quad \displaystyle \int _1^e x^2 \ln x\, dx & \quad \text{b)} \quad \displaystyle \int _0^1 x^2 e^x\, dx \end{array} \]

3.4 Integration of Trigonometric and Hyperbolic Functions

The videos in this section demonstrate how to integrate trigonometric, inverse trigonometric, and hyperbolic functions using various techniques, including identities and standard formulas.

• i)

  Integration using trigonometric identities

  \[ \begin{array}{lll} \text{a)}\quad \displaystyle \int \frac{3}{7} \sin x\, dx & \quad \text{b)}\quad \displaystyle \int \frac{4 \sec ^2 x}{5}\, dx & \quad \text{c)}\quad \displaystyle \int \frac{5}{\sec x}\, dx \\[1.2em] \text{d)}\quad \displaystyle \int \frac{2}{7} \sin x\, dx & \quad \text{e)}\quad \displaystyle \int \frac{3}{8 \csc x}\, dx & \quad \text{f)}\quad \displaystyle \int \frac{-4 \cos x}{7}\, dx \\[1.2em] \text{g)}\quad \displaystyle \int \left(3x - \frac{\sec ^2 x}{8}\right) dx & \quad \text{h)}\quad \displaystyle \int \left(1 + \tan ^2 x\right) dx & \end{array} \]

  \[ \begin{array}{lll} \text{a)}\quad \displaystyle \int \frac{3}{4} \sin (5x - 2)\, dx & \quad \text{b)}\quad \displaystyle \int \frac{4 \sec ^2(2 - 3x)}{5}\, dx & \quad \text{c)}\quad \displaystyle \int \frac{3}{\sec 2x}\, dx \\[1.2em] \text{d)}\quad \displaystyle \int 5 \cos (4x - 7)\, dx & \quad \text{e)}\quad \displaystyle \int \frac{2 \sin (3 - 8x)}{7}\, dx & \quad \text{f)}\quad \displaystyle \int \frac{3}{\cos ^2(2x - 5)}\, dx \end{array} \]

  \[ \begin{array}{lll} \text{a)}\quad \displaystyle \int \left(1 + \tan ^2 x \right)\, dx & \quad \text{b)}\quad \displaystyle \int \left(1 + \tan ^2(5\theta ) \right)\, d\theta & \quad \displaystyle \int \left(3 + 3 \tan ^2 2x \right)\, dx \end{array} \]

  \[ \begin{array}{ll} \text{a)}\quad \displaystyle \int \sin x \cos x \, dx & \quad \text{b)}\quad \displaystyle \int 5 \cos \left(\frac{3}{2}x\right) \sin \left(\frac{3}{2}x\right) \, dx \end{array} \]

  \[ \begin{array}{ll} \text{a)}\quad \displaystyle \int \sin ^2 x \, dx & \quad \quad \qquad \text{b)}\quad \displaystyle \int 3 \sin ^2(5\theta ) \, d\theta \end{array} \]

• ii)

  Integration using inverse trigonometric functions

  a)  \(\displaystyle \int \frac{2}{\sqrt{16 - x^2}} \, dx\)	b)  \(\displaystyle \int \frac{dx}{5 + 16x^2}\)
  c)  \(\displaystyle \int \frac{dx}{x\sqrt{9x^2 - 4}}\)

• iii)

  Integration by subsitution of hyperbolic functions

  Set 1

  a) \(\displaystyle \int \frac{1}{\sqrt{x^2 + 4}} \, dx\)

  b) \(\displaystyle \int \frac{1}{\sqrt{x^2 - 9}} \, dx\)

  c) \(\displaystyle \int \frac{1}{25 - x^2} \, dx\)

  Set 2

  a) \(\displaystyle \int \frac{1}{\sqrt{3x^2 + 27}} \, dx\)	b) \(\displaystyle \int \frac{1}{\sqrt{x^2 - 6x}} \, dx\)
  c) \(\displaystyle \int \frac{1}{\sqrt{7 - 6x - x^2}} \, dx\)	d) \(\displaystyle \int \frac{1}{\sqrt{12x + 2x^2}} \, dx\)

  Set 3

  a) \(\displaystyle \int \sqrt{1 + x^2} \, dx\)

  b) \(\displaystyle \int _0^6 \frac{x^3}{\sqrt{x^2 + 9}} \, dx\)

  c) \(\displaystyle \int \frac{1}{\sqrt{4x^2 - 12x - 7}} \, dx\)

3.5 Reduction Formulae

The following link demonstrates how to use the reduction formula method to evaluate the integrals listed below.

3.6 Integration by using partial fractions and completing the square

• i)

  The link below demonstrates how to find the partial fractions for the following improper rational functions, but not the integral.

  a) \(\displaystyle \frac{5x - 3}{x^2 - 3x - 4}\)	b) \(\displaystyle \frac{6x - 22}{2x^2 + 7x - 15}\)	c) \(\displaystyle \frac{7x - 11}{(x - 2)^2}\)
  d) \(\displaystyle \frac{3x^2 - 24x + 53}{x^3 - 6x^2 + 9x}\)	e) \(\displaystyle \frac{6x^2 + 21x + 11}{(x^2 + 3)(x + 5)}\)	f) \(\displaystyle \frac{3x^2 + 5x - 4}{(x^2 - 7)(x + 1)}\)
  g) \(\displaystyle \frac{3x^4 - 2x^3 + 6x^2 - 3x + 3}{(x^2 + 2)^2(x + 3)}\)	h) \(\displaystyle \frac{x^3 + 3}{x^2 - 2x - 3}\)

• ii)

  The link below demonstrates how to evaluate integrals of the following rational functions by using partial fractions.

  a)  \(\displaystyle \int \frac{1}{x^2 - 4} \, dx\)	b)  \(\displaystyle \int \frac{x - 4}{x^2 + 2x - 15} \, dx\)
  c)  \(\displaystyle \int \frac{x}{(x - 1)(x - 2)^2} \, dx\)	d)  \(\displaystyle \int \frac{x^2 + 9}{(x^2 - 1)(x^2 + 4)} \, dx\)

• iii)

  The video in the link below demonstrates how to evaluate integrals using the completing the square method for the following questions.

  a)  \(\displaystyle \int \frac{dx}{x^2 - 6x + 13}\)

  b)  \(\displaystyle \int \frac{x - 5}{x^2 + 8x + 22} \, dx\)