Antiderivatives and Integrals
Instructions
- First, you should watch the concept videos below explaining the topics in the section.
- Next, you should attempt to solve the exercises. You can check your answers and watch videos explaining the exercises.
- When you have finished the material below, you can start on the next section or return to the main workshop page.
Concepts
- Finding an antiderivative
- Solving an indefinite integral
- Finding a function from its derivative and a given function value
Exercises
Directions: You should attempt to solve the problems first and then watch the video to see the solution.
1. Find the most general antiderivative of the following function.
\[H'(x)=\frac{7}{x}+\frac{2}{5}\]
\[H'(x)=\frac{7}{x}+\frac{2}{5}\]
2. Find the following indefinite integral
\[\int \! \left( e^x-13x^{-2.4}\right) \, dx\]
\[\int \! \left( e^x-13x^{-2.4}\right) \, dx\]
3. Find the following indefinite integral.
\[ \int \frac{7^x\sqrt[3]{x}+x^{-\frac{2}{3}}}{x^{\frac{1}{3}}} \, dx\]
\[ \int \frac{7^x\sqrt[3]{x}+x^{-\frac{2}{3}}}{x^{\frac{1}{3}}} \, dx\]
4. Find \(f(x)\) if \(f'(x)=8x^3-12x^2+13\) and \(f(-1)=18.\)
5. The pastry shop Beignet and the Jets has found its daily marginal cost function to be \(C'(x)=3-0.06x\) dollars per beignet when \(x\) beignets are made each day. If it costs the pastry shop $65 each day to make 20 beignets, how much does it cost to make 40 beignets each day?