Activity 4.4.3.
Use your knowledge of derivatives of basic functions to complete Table 4.4.5 of antiderivatives. For each entry, your task is to find a function \(F\) whose derivative is the given function \(f\text{.}\) When finished, use the FTC and the results in the table to evaluate the three given definite integrals.
| given function, \(f(x)\) | antiderivative, \(F(x)\) |
| \(k\text{,}\) (\(k\) is constant) | |
| \(x^n\text{,}\) \(n \ne -1\) | |
| \(\frac{1}{x}\text{,}\) \(x \gt 0\) | |
| \(\sin(x)\) | |
| \(\cos(x)\) | |
| \(\sec(x) \tan(x)\) | |
| \(\csc(x) \cot(x)\) | |
| \(\sec^2 (x)\) | |
| \(\csc^2 (x)\) | |
| \(e^x\) | |
| \(a^x\) \((a \gt 1)\) | |
| \(\frac{1}{1+x^2}\) | |
| \(\frac{1}{\sqrt{1-x^2}}\) |
(a)
\(\displaystyle \int_0^1 \left(x^3 - x - e^x + 2\right) \,dx\)
(b)
\(\displaystyle \int_0^{\pi/3} (2\sin (t) - 4\cos(t) + \sec^2(t) - \pi) \, dt\)
(c)
\(\displaystyle \int_0^1 (\sqrt{x}-x^2) \, dx\)
