Table - basic integrals
Task number: 3105
Complete the following table:
| f(x) | F(x) | interval |
|---|---|---|
| \(x^a, r\in \mathbb R \) {-1} | ||
| \(\frac {1}{x}\) | ||
| \(e^x\) | ||
| sin(\(x\)) | ||
| cos(\(x\)) | ||
| \(\frac {1}{\text {cos}^2x}\) | ||
| \(\frac {1}{1+x^2}\) | ||
| \(\frac {1}{\sqrt {1-x^2}}\) |
Resolution
f(x) F(x) interval \(x^a, r\in \mathbb R\) \ {-1} \(\frac {1}{a+1}x^{a+1}\) \(\mathbb R\) for \(a \in \mathbb N\) ∪ {0}
(-∞, 0) and (0, ∞) for \(a \in \mathbb N\)
(0, ∞) for \(a \in \mathbb R\) \ \(\mathbb Z\)\(\frac {1}{x}\) ln(|\(x\)|) (-∞, 0) and (0, ∞) \(e^x\) \(e^x\) \(\mathbb R\) sin(\(x\)) -cos(\(x\)) \(\mathbb R\) cos(\(x\)) sin(\(x\)) \(\mathbb R\) \(\frac {1}{\text {cos}^2x}\) tan(\(x\)) \((-\pi/2 + k\pi, \pi/2 + k\pi), k \in \mathbb Z\) \(\frac {1}{1+x^2}\) arctan(\(x\)) \(\mathbb R\) \(\frac {1}{\sqrt {1-x^2}}\) arcsin(\(x\)) (-1, 1)
