Find an antiderivative of \(7\).

What term must be included in \(\int f(x)\,dx\) to represent all antiderivatives?

Complete the statement: if \(F'(x)=f(x)\), then \(\int f(x)\,dx=\,?\)

Why is \(C\) needed in \(\int 3x^2\,dx=x^3+C\)?

Find \(\int x^4\,dx\).

Find \(\int 6x^2\,dx\).

Find \(\int (4x^3-5)\,dx\).

Find \(\int \frac{1}{x^3}\,dx\) for \(x\ne0\).

Find \(\int (3x^2-4x+8)\,dx\).

Find \(\int \left(2\sqrt{x}+\frac{3}{x}\right)dx\) for \(x>0\).

Find \(F(x)\) if \(F'(x)=5x^4-2x\) and \(F(1)=10\).

Verify that \(F(x)=\frac{x^5}{5}-3x+C\) is an antiderivative of \(x^4-3\).

Find \(\int \left(\frac{2}{\sqrt{x}}-x^{-2}\right)dx\) for \(x>0\).

Find \(F(x)\) if \(F''(x)=12x-6\), \(F'(0)=5\), and \(F(0)=2\).

Find \(a\) if \(\int (ax^2+4)\,dx=2x^3+4x+C\).

For which \(n\) does the power rule formula \(\int x^n\,dx=\frac{x^{n+1}}{n+1}+C\) fail, and what integral replaces it?

Find \(F(x)\) if \(F'(x)=2x+k\), \(F(0)=3\), and \(F(2)=15\).

A student writes \(\int x^{-1}\,dx=\frac{x^0}{0}+C\). Explain the error and give the correct antiderivative.

Show that any two antiderivatives of the same function differ by a constant, assuming the fact that a function with derivative zero on an interval is constant.

Find all functions \(F\) such that \(F'(x)=6x^2-4x+1\) and \(F(1)-F(0)=5\), or explain if none exist.