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 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 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
Exercises: Use the table of integrals and the properties above to evaluate the following integrals. [Note that you may need to use more than one of the above properties for one integral].
More references on integrals and their applications in calculus. |
