Definite Integrals
Watch this video to learn about evaluating definite integrals.
Definite integrals are integrals which have limits (upper and lower) and can be evaluated to give a definite answer.
A question of this type may look like:
\(\int\limits_a^b {a{x^n}\,\,dx} = \left[ {\frac{{a{x^{n + 1}}}}{{n + 1}}} \right]_a^b\)
Example
\(\int\limits_1^2 {{x^4}\,\,dx}\)
Solution
\(\int\limits_1^2 {{x^4}\,\,dx}\)
\(= \left[ {\frac{{{x^5}}}{5}} \right]_1^2\)
Substitute the limits in and subtract.
\(= \left( {\frac{{{2^5}}}{5}} \right) - \left( {\frac{{{1^5}}}{5}} \right)\)
\(= \frac{{31}}{5}\)
Example 2 (extension)
\(\int\limits_1^{\sqrt 3 }x\,\,dx\)
Solution
\(\int\limits_1^{\sqrt 3 }x\,\,dx\)
\(= \left[ {\frac{{{x^2}}}{2}} \right]_1^{\sqrt 3 }\)
\(= \left( {\frac{{{{(\sqrt 3 )}^2}}}{2}} \right) - \left( {\frac{{{1^2}}}{2}} \right)\)
\(= \frac{3}{2} - \frac{1}{2}\)
\(= 1\)
Example 3 (extension)
\(\int\limits_0^{\frac{\pi }{2}} {\sin (3x + \frac{\pi }{4})}\,\, dx\)
Solution
\(\int\limits_0^{\frac{\pi }{2}} {\sin (3x + \frac{\pi }{4})}\,\, dx\)
\(= \left[ {\frac{{ - 1}}{3}\cos (3x + \frac{\pi }{4})} \right]_0^{\frac{\pi }{2}}\)
\(= \left( {\frac{{ - 1}}{3}\cos \left( {3\left( {\frac{\pi }{2}} \right) + \frac{\pi }{4}} \right)} \right) - \left( {\frac{{ - 1}}{3}\cos \left( {3(0) + \frac{\pi }{4}} \right)} \right)\)
\(= \left( {\frac{{ - 1}}{3}\cos \left( {\frac{{7\pi }}{4}} \right)} \right) - \left( {\frac{{ - 1}}{3}\cos \left( {\frac{\pi }{4}} \right)} \right)\)
\(= - \frac{{\sqrt 2 }}{6} - \left( { - \frac{{\sqrt 2 }}{6}} \right)\)
\(= 0\)
