Activity 4.4.3 — Active Calculus

Use your knowledge of derivatives of basic functions to complete Table of antiderivatives. For each entry, your task is to find a function $F$ whose derivative is the given function $f$ . When finished, use the FTC and the results in the table to evaluate the three given definite integrals.

Familiar basic functions and their antiderivatives.
given function, $f(x)$	antiderivative, $F(x)$
$k$ , ( $k$ is constant)
$x^n$ , $n \ne -1$
$\frac{1}{x}$ , $x \gt 0$
$\sin(x)$
$\cos(x)$
$\sec(x) \tan(x)$
$\csc(x) \cot(x)$
$\sec^2 (x)$
$\csc^2 (x)$
$e^x$
$a^x$ $(a \gt 1)$
$\frac{1}{1+x^2}$
$\frac{1}{\sqrt{1-x^2}}$

$\displaystyle \int_0^1 \left(x^3 - x - e^x + 2\right) \,dx$
$\displaystyle \int_0^{\pi/3} (2\sin (t) - 4\cos(t) + \sec^2(t) - \pi) \, dt$
$\displaystyle \int_0^1 (\sqrt{x}-x^2) \, dx$

Boelkins, Active Calculus, §4.4 The Fundamental Theorem of Calculus, Activity 4.4.3

AI Tutor