Definite integrals

In a nutshell

Definite integrals are integrals that do not have a constant of integration, and instead give a numerical answer.

Definition

A definite integral is an integral of the form:

Evaluating definite integrals

To evaluate definite integrals, follow this procedure.

This can also be written as:

​$$\int_b^af'(x) \,dx=\left[f(x)\right]_b^a=f(a)-f(b)$$​​

Example 1

Evaluate the definite integral $$\int_2^5\dfrac{1+x}{x^3}\,dx$$.

Write down each term in the form $$ax^n$$:

​$$\begin{aligned}\int_2^5\dfrac{1+x}{x^3}\,dx&=\int_2^5\left(\dfrac{1}{x^3}+\dfrac{x}{x^3}\right) \,dx\\&=\int_2^5(x^{-3}\,+x^{-2})\,dx\end{aligned}$$​​

Integrate the terms, writing the integral in square brackets:

​$$\begin{aligned}\int_2^5(x^{-3}\,+x^{-2})\,dx&=\left[\dfrac{1}{-2}x^{-2}+\dfrac{1}{-1}x^{-1}\right]_2^5\\&=\left[-\dfrac{1}{2x^2}-\dfrac{1}{x}\right]_2^5\end{aligned}$$​​

Evaluate the upper and lower limits and subtract them from each other:

$$\begin{aligned}\left[-\dfrac{1}{2x^2}-\dfrac{1}{x}\right]_2^5 &= \left(-\dfrac{1}{2(5)^2}-\dfrac{1}{5}\right)-\left(-\dfrac{1}{2(2)^2}-\dfrac{1}{2}\right)\\&=\left(-\dfrac{1}{50}-\dfrac{1}{5}\right)-\left(-\dfrac{1}{8}-\dfrac{1}{2}\right)\\&=\left(-\dfrac{11}{50}\right)-\left(-\dfrac{5}{8}\right)\\&=\dfrac{5}{8}-\dfrac{11}{50}\\&=\dfrac{81}{200}\end{aligned}$$​​

$$\underline{\int_2^5\dfrac{1+x}{x^3}\,dx=\dfrac{81}{200}}$$​

Note: The order of subtraction matters! Always make sure to subtract the lower limit from the upper limit.