Table - basic integrals

Task number: 3105

Complete the following table:

f(x) F(x) interval
\(x^a, r\in \mathbb R \) {-1}
\(\frac {1}{x}\)
\(\frac {1}{\text {cos}^2x}\)
\(\frac {1}{1+x^2}\)
\(\frac {1}{\sqrt {1-x^2}}\)
  • 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)
Difficulty level: Easy task (using definitions and simple reasoning)
Routine calculation training
Cs translation
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