Evaluation of Definite Integrals | Engineering Mathematics for Mechanical Engineering PDF Download

Definite Integration

The integral is called definite integral and is defined to be  where ϕ(x) is the indefinite integral of ∫f(x)dx  . It is read as integral from a to b of f(x), a is called the lower limit and b is called the upper limit. This is also called the theorem of calculus.

Properties of definite integral

Improper Integrals

• The integral  is said to be an improper integral of the first kind, if one or both the limits of integration are infinite.
• The integral  is said to be improper integral of second kind, if f(x) becomes unbounded at one or more points in the interval of integration [a, b].

Limit comparison Test for Convergence and Divergence of Improper Integrals : 

If f(x) and g(x) are positive functions and if
 both converge or both diverge.

Solved Numericals

Q1. Integrate .
Solution: We know that ∫cosxdx = sinx + c
∴

Q2. Integrate.
Solution: We know that

Q3. Evaluate .
Solution: We know that

Q4. Evaluate .
Solution: Let

Q5. Solve .
Solution: The integrand becomes infinite at x = 1.
The area under the curve between the lines x = 0 and x = 1 is not well defined since the curve extends to infinity as x tends to 1 from the left. Nevertheless, we can define the area from x = 0 to x = t where 0 < t < 1.
Then the integral  
We also say that the improper integral converges.