Basic Integration Formulas
Recall the integration formulas given in the table in Antiderivatives and the rule on properties of definite integrals. Let’s look at a few examples of how to apply these rules.
Example 5.23
Integrating a Function Using the Power Rule
Use the power rule to integrate the function \(\int_{1}^{4} \sqrt{t} \left(\right. 1 + t \left.\right) d t .\)
Solution
The first step is to rewrite the function and simplify it so we can apply the power rule:
\[\begin{aligned} \int_{1}^{4} \sqrt{t} \left(\right. 1 + t \left.\right) d t & = \int_{1}^{4} t^{1 / 2} \left(\right. 1 + t \left.\right) d t \\ \\ & = \int_{1}^{4} \left(\right. t^{1 / 2} + t^{3 / 2} \left.\right) d t . \end{aligned}\]
Now apply the power rule:
\[\begin{aligned} \int_{1}^{4} \left(\right. t^{1 / 2} + t^{3 / 2} \left.\right) d t & = \left(\left(\right. \frac{2}{3} t^{3 / 2} + \frac{2}{5} t^{5 / 2} \left.\right) \left|\right.\right)_{1}^{4} \\ & = \left[\right. \frac{2}{3} \left(\left(\right. 4 \left.\right)\right)^{3 / 2} + \frac{2}{5} \left(\left(\right. 4 \left.\right)\right)^{5 / 2} \left]\right. - \left[\right. \frac{2}{3} \left(\left(\right. 1 \left.\right)\right)^{3 / 2} + \frac{2}{5} \left(\left(\right. 1 \left.\right)\right)^{5 / 2} \left]\right. \\ & = \frac{256}{15} . \end{aligned}\]
Checkpoint 5.21
Find the definite integral of \(f \left(\right. x \left.\right) = x^{2} - 3 x\) over the interval \(\left[\right. 1 , 3 \left]\right. .\)
This lesson is part of:
Integration
View Full Tutorial