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 contant value.

1 - Integral of k f(x).


   
k f(x) dx = k f(x) dx

Example: Evaluate the integral

5 sinx dx


Solution:

According to the above property

5 sinx dx = 5sin(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.


   
[f(x) + g(x)] dx = f(x) dx + g(x) dx

Example: Evaluate the integral

[x + e x] dx


Solution:

According to the above property

[x + e x] dx = x dx + e x dx

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.


   
[f(x) - g(x)] dx = f(x) dx - g(x) dx

Example: Evaluate the integral

[2 - 1/x] dx


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.


   
[f(u) du/dx] dx = f(u) du

Example: Evaluate the integral

(x 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.


   
f(x) g '(x) dx = f(x) g(x) - f '(x) g(x) dx

Example: Evaluate the integral

x cos x dx


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].

1. (1/2) ln x dx

2. [sin x + x 5] dx

3. [sinh x - 3] dx

4. -x sin x dx



Answers to Above Exercises

1. [x ln x] / 2 - x / 2

2. x 6 / 6 - cos x

3. cosh x - 3x

4. x cos x - sin x


More references on
integrals and their applications in calculus.






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