# Properties of Integrals - Tutorial

A tutorial, with examples and detailed solutions, in using the properties of indefinite integrals in calculus is presented. A set of exercises with answers is presented after the tutorial.

In what follows, C is a constant of integration and can take any constant value.
## 1 - Integral of k f(x).Example: Evaluate the integral
Solution:According to the above property ∫5 sinx dx = 5∫sin(x) dx ∫sin(x) dx is given by 2.1 in table of integrals, hence ∫5 sin(x) dx = -5cos x + C
## 2 - Integral of Sum of Functions.Example: Evaluate the integral
^{ x}] dx
Solution:According to the above property ∫[x + e ^{ x}] dx = ∫x dx + ∫e^{ x} dx
ò x dx is given by 1.3 and ∫e ^{ x} dx
by 4.1 in table of integrals, hence
∫[x + e ^{ x}] dx = x^{ 2} / 2 + e^{ x} + c
## 3 - Integral of Difference of Functions.Example: Evaluate the integral
Solution:According to the above property ∫[2 - 1/x] dx = ∫2 dx - ∫(1/x) dx ∫2 dx is given by 1.2 and ∫(1/x) dx by 1.4 in table of integrals, hence ∫[2 - 1/x] dx = 2x - ln |x| + c
## 3 - Integration by Substitution.Example: Evaluate the integral
^{ 2} - 1)^{ 20} 2x dx
Solution:Let u = x ^{ 2} - 1, du/dx = 2x and the given integral can be written as
∫(x ^{ 2} - 1)^{ 20} 2x dx
= ∫u ^{ 20} (du/dx) dx
= ∫u ^{ 20} du according to above property
= u ^{ 21} / 21 + c = (x^{ 2} - 1)^{ 21} / 21 + c
## 3 - Integration by Parts.Example: Evaluate the integral
Solution:Let f(x) = x and g ' (x) = cos x which gives f ' (x) = 1 and g(x) = sin x From integration by parts formula above, ∫x cos x dx = x sin x - ∫1 sin x dx = x sin x + cos x + c
More references on integrals and their applications in calculus. |