Use the standard table to find \(\int e^x\,dx\).

Use the standard table to find \(\int \cos x\,dx\).

Find \(\int \sin x\,dx\).

Find \(\int \sec^2 x\,dx\).

Find \(\int x^7\,dx\).

Find \(\int 5^x\,dx\).

Find \(\int \frac{1}{1+x^2}\,dx\).

Find \(\int \frac{1}{\sqrt{1-x^2}}\,dx\).

Find \(\int (3x^2+2\cos x-e^x)\,dx\).

Find \(\int \left(\frac{4}{x}+\sec^2 x\right)dx\) for \(x\ne0\).

Find \(\int (2^x+\sin x)\,dx\), and check the exponential term.

Find \(\int (\csc^2 x+\sec x\tan x)\,dx\).

Find \(\int \left(x^{-3}+\frac{1}{x}+e^x\right)dx\) for \(x\ne0\).

Find \(\int \left(\frac{1}{\sqrt{1-x^2}}-\frac{1}{1+x^2}\right)dx\).

Find \(k\) if \(\int k^x\,dx=\frac{k^x}{\ln 3}+C\) for all \(x\), with \(k>0\) and \(k\ne1\).

Choose a substitution type from the standard trigonometric substitution table for \(\sqrt{9-x^2}\).

Choose a substitution type from the standard trigonometric substitution table for \(\sqrt{x^2-16}\).

A student writes \(\int \frac1x\,dx=\frac{1}{2}x^2+C\). Explain the mistake and give the correct standard integral.

A student claims \(\int 4^x\,dx=4^x+C\). Diagnose the error and correct the answer.

Explain why \(\int \frac{1}{|x|\sqrt{x^2-1}}\,dx=\operatorname{arcsec}|x|+C\) requires a domain restriction.