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Definite integrals

Definite integrals

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Tutor: Bilal

Summary

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:


∫baf(x) dx\int_b^a f(x)\,dx∫ba​f(x)dx​​

​aaa​​

The upper limit of the integral.

​bbb​​

The lower limit of the integral.



Evaluating definite integrals

To evaluate definite integrals, follow this procedure.


1.

Evaluate the integral without the constant of integration.

2.

Write down the integral between square brackets, and write the lower and upper limits at the bottom right and top right of the brackets respectively.

3.

Substitute the upper and lower limits into the integral. Use round brackets here.

4.

Subtract these two numbers from each other.


This can also be written as:

​∫baf′(x) dx=[f(x)]ba=f(a)−f(b)\int_b^af'(x) \,dx=\left[f(x)\right]_b^a=f(a)-f(b)∫ba​f′(x)dx=[f(x)]ba​=f(a)−f(b)​​


Example 1

Evaluate the definite integral ∫251+xx3 dx\int_2^5\dfrac{1+x}{x^3}\,dx∫25​x31+x​dx.


Write down each term in the form axnax^naxn:

​∫251+xx3 dx=∫25(1x3+xx3) dx=∫25(x−3 +x−2) dx\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}∫25​x31+x​dx​=∫25​(x31​+x3x​)dx=∫25​(x−3+x−2)dx​​​


Integrate the terms, writing the integral in square brackets:

​∫25(x−3 +x−2) dx=[1−2x−2+1−1x−1]25=[−12x2−1x]25\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}∫25​(x−3+x−2)dx​=[−21​x−2+−11​x−1]25​=[−2x21​−x1​]25​​​​


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

[−12x2−1x]25=(−12(5)2−15)−(−12(2)2−12)=(−150−15)−(−18−12)=(−1150)−(−58)=58−1150=81200\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}[−2x21​−x1​]25​​=(−2(5)21​−51​)−(−2(2)21​−21​)=(−501​−51​)−(−81​−21​)=(−5011​)−(−85​)=85​−5011​=20081​​​​


∫251+xx3 dx=81200‾\underline{\int_2^5\dfrac{1+x}{x^3}\,dx=\dfrac{81}{200}}∫25​x31+x​dx=20081​​​


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

 



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FAQs - Frequently Asked Questions

How do you evaluate definite integrals?

First, integrate the function, then substitute the lower and upper limits into the integral and subtract the evaluated lower limit from the evaluated upper limit.

What is definite integration?

Definite integration is integration with limits. It has no constant of integration and it gives a numerical answer.

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