Unit 8: Definite Integrals

As introduced in Unit 1, indefinite and definite integrals are two core concepts in calculus. An indefinite integral (or antiderivative) represents a general form of accumulation without specified bounds, producing a function that describes accumulated change. In contrast, a definite integral evaluates this accumulation between two specific points on the x-axis, yielding a real number that quantifies the total change or area over the interval.

The Fundamental Theorem of Calculus connects the concepts of indefinite and definite integrals by demonstrating that the definite integral of a function [latex]f(x)[/latex] over an interval [latex][a, b][/latex] can be evaluated using an antiderivative [latex]F(x)[/latex] of [latex]f(x)[/latex], where [latex]F'(x)=f(x)[/latex]. In mathematical terms, the theorem is expressed as:

[latex]\int_a^b f(x) dx= F(b)-F(a)[/latex]

This theorem shows that a definite integral can be computed by simply finding the difference in the values of an antiderivative at the upper limit, [latex]b[/latex], and lower limit, [latex]a[/latex].

Having introduced various techniques for evaluating indefinite integrals, let’s now discuss how to apply these methods to definite integrals. When working with definite integrals, we use the same integration techniques—such as substitution, integration by parts, and partial fraction decomposition—but with the addition of evaluating limits.

Practice Questions:

Evaluate the following indefinite integrals before referring to the video for comprehensive solutions.

Q1. [latex]\int_0^3 4^x + x^4 \; dx[/latex]   (similar to Q1 in Unit 3)

Q2. [latex]\int_{-1}^1 e^x + x^e + e \; dx[/latex]   (similar to Q2 in Unit 3)

Q3. [latex]\int_1^e 3 \ln x - 5 x^{\pi^2} \;dx[/latex]    (similar to Q4 in Unit 3)

Q4.  [latex]\int_4^9 \frac{\sqrt{x}}{5}- \frac{5}{\sqrt{x}}\; dx[/latex]    (similar to Q6 in Unit 3)

Q5. [latex]\int_0^{\pi/4} 10\cos t  + 3\tan t  \;dt[/latex]    (similar to Q10 in Unit 3)

Q3. [latex]\int x^2e^x\;dx[/latex]   (similar to Q3 in Unit 6, Partial Fractions method)