Evaluation of Definite Integrals

Integration by substitution is a method in which a suitable part of the integrand is replaced by a new variable so that the integral becomes easier to evaluate. 

In definite integrals, after substitution we may either resubstitute back to the original variable and use the original limits, or change the limits according to the new variable and evaluate directly.

• Consider the integral temporarily without limits and make a suitable substitution t = g(x).

• Find the antiderivative and resubstitute the new variable back into terms of the original variable x.

• Evaluate the resulting expression using the original integration limits (a and b).

• Substitute a new variable, such as t = g(x), meaning \[dt = g'(x) \, dx\].

• Change the limits of integration to match the new variable:

  • Lower limit becomes \[t_{lower} = g(a)\]

  • Upper limit becomes \[t_{upper} = g(b)\]

• Integrate the new integrand with respect to t and evaluate it directly using the new limits—no need to substitute back to x.

Substitution with Inverse Trigonometric Functions

Evaluate:

Solution:

Let \[t = \tan^{-1}x\], then \[dt = \frac{1}{1 + x^2} \, dx\].

Update the limits:

• When x = 0, \[t = \tan^{-1}(0) = 0\]
• When x = 1, \[t = \tan^{-1}(1) = \frac{\pi}{4}\]

Substitute into the integral:

• Look for an inner function and its derivative.

• Choose substitution carefully.

• If the integral is continued in the new variable, change the limits accordingly.

• Integrate in the new variable.

• Do not add +C in a definite integral.

• Once the limits have been changed to the new variable, there is no need to substitute back to the original variable.