Basic antiderivatives and substitution

Understand the idea

Antidifferentiation reverses derivative rules and includes an arbitrary constant. The power rule for integration excludes exponent $\displaystyle -1$ , which produces a logarithm. Substitution matches a composite expression with its derivative. For definite integrals either change both bounds into the new variable or substitute back before evaluating; never mix transformed variables with original bounds. Other useful reverse patterns are $\displaystyle \int\csc^2x\,dx=-\cot x+C$ and $\displaystyle \int\csc x\cot x\,dx=-\csc x+C$ . For an inverse-sine pattern, $\displaystyle \int dx/\sqrt{a^2-x^2}=\arcsin(x/a)+C$ when $\displaystyle a>0$ and $\displaystyle |x|<a$ .

\begin{aligned}\int x^n\,dx&=\frac{x^{n+1}}{n+1}+C\quad(n\ne-1)\\\int\frac{u'}{u}\,dx&=\ln|u|+C\\\int e^x\,dx&=e^x+C\\\int\sin x\,dx&=-\cos x+C\\\int\cos x\,dx&=\sin x+C\\\int\sec^2x\,dx&=\tan x+C\\\int\sec x\tan x\,dx&=\sec x+C\end{aligned}

Worked example

Evaluate $\displaystyle \int_0^1 2x e^{x^2}\,dx$ .

Let

\displaystyle u=x^2

, so

\displaystyle du=2x\,dx

. The bounds become

\displaystyle 0

and

\displaystyle 1

. The integral is

\displaystyle \int_0^1e^u\,du=e-1

. Differentiation of

\displaystyle e^{x^2}

checks the antiderivative.

Try it yourself

Find $\displaystyle \int 3x^2/(x^3+5)\,dx$ .

Show solution and reasoning

With $\displaystyle u=x^3+5$ , the result is $\displaystyle \ln|x^3+5|+C$ on any interval avoiding the denominator zero.