Integrals
Integral
The definite integral of a function \(f(x)\) over the interval \([a, b]\) is:
\[ \int_a^b f(x) \, dx \]
This represents the signed area under the curve of \(f(x)\) from \(a\) to \(b\).
The indefinite integral (or antiderivative) is:
\[ \int f(x) \, dx = F(x) + C \]
where \(F'(x) = f(x)\) and \(C\) is an arbitrary constant of integration.
Integration Techniques
Table of Integration Techniques.
| Rule | Function \(f(x)\) | Integral \(\int f(x)\,dx\) | Notes |
|---|---|---|---|
| Constant | \(c\) | \(cx + C\) | \(c \in \mathbb{R}\) |
| Power | \(x^n\) (with \(n \neq -1\)) | \(\frac{x^{n+1}}{n+1} + C\) | Power rule for \(n \neq -1\) |
| Sum/Difference | \(g(x) \pm h(x)\) | \(\int g(x)\,dx \pm \int h(x)\,dx\) | Integral distributes over sums and differences |
| Exponential (base e) | \(e^x\) | \(e^x + C\) | Natural exponential |
| Exponential (base a) | \(a^x\) | \(\frac{a^x}{\ln a} + C\) | \(a > 0,\ a \neq 1\) |
| Logarithmic | \(\frac{1}{x}\) | \(\ln|x| + C\) | Valid for \(x \neq 0\) |
| Chain Rule (u-sub) | \(f(g(x))g'(x)\) | \(\int f(u)\,du\), where \(u = g(x)\) | Substitution method |
| Integration by Parts | \(u(x)v'(x)\) | \(u(x)v(x) - \int u'(x)v(x)\,dx\) | \(\int u\,dv = uv - \int v\,du\) |