Evaluation of Definite Integrals by Substitution

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, the limits must also be changed according to the new variable.

• Ignore the integration limits initially and integrate the function using 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.

• Change the limits immediately.

• Integrate in the new variable.

• Do not add +C in a definite integral.