Integration reverses differentiation: given a rate of change, it recovers the total. This chapter develops the main techniques – standard results, substitution, special forms, partial fractions and integration by parts – and then the definite integral, which measures accumulated quantities such as area. The Fundamental Theorem of Calculus connects the two ideas: \(\int_a^b f(x)\,dx=F(b)-F(a)\) for any antiderivative \(F\).
| \(f(x)\) | \(\int f(x)\,dx\) |
|---|---|
| \(x^n\ (n\neq-1)\) | \(\frac{x^{n+1}}{n+1}+C\) |
| \(\frac1x\) | \(\log|x|+C\) |
| \(e^x\) | \(e^x+C\) |
| \(a^x\) | \(\frac{a^x}{\log a}+C\) |
| \(\sin x\) | \(-\cos x+C\) |
| \(\cos x\) | \(\sin x+C\) |
| \(\sec^2x\) | \(\tan x+C\) |
| \(\csc^2x\) | \(-\cot x+C\) |
| \(\sec x\tan x\) | \(\sec x+C\) |
| \(\frac{1}{\sqrt{1-x^2}}\) | \(\sin^{-1}x+C\) |
| \(\frac1{1+x^2}\) | \(\tan^{-1}x+C\) |
\[\int\tan x=\log|\sec x|,\ \int\cot x=\log|\sin x|,\ \int\sec x=\log|\sec x+\tan x|,\ \int\csc x=\log|\csc x-\cot x|\]
| Integral | Result |
|---|---|
| \(\int\frac{dx}{x^2-a^2}\) | \(\frac1{2a}\log\left|\frac{x-a}{x+a}\right|+C\) |
| \(\int\frac{dx}{a^2-x^2}\) | \(\frac1{2a}\log\left|\frac{a+x}{a-x}\right|+C\) |
| \(\int\frac{dx}{x^2+a^2}\) | \(\frac1a\tan^{-1}\frac xa+C\) |
| \(\int\frac{dx}{\sqrt{x^2-a^2}}\) | \(\log\left|x+\sqrt{x^2-a^2}\right|+C\) |
| \(\int\frac{dx}{\sqrt{a^2-x^2}}\) | \(\sin^{-1}\frac xa+C\) |
| \(\int\frac{dx}{\sqrt{x^2+a^2}}\) | \(\log\left|x+\sqrt{x^2+a^2}\right|+C\) |
\[\int uv\,dx=u\int v\,dx-\int\left(u'\int v\,dx\right)dx\]
ILATE: Inverse trig, Log, Algebraic, Trig, Exponential.
\[\int e^x\left[f(x)+f\,'(x)\right]dx=e^xf(x)+C\]
\(\int_0^af(x)dx=\int_0^af(a-x)dx\); \(\int_{-a}^af=2\int_0^af\) (even) or 0 (odd).
\(\int\frac{f\,'(x)}{f(x)}dx=\log|f(x)|+C\), e.g. \(\int\frac{2x+3}{x^2+3x+1}dx=\log|x^2+3x+1|+C\).
In \(\int_0^{\pi/2}\sin^2x\cos x\,dx\) put \(u=\sin x\): limits become 0 and 1, giving \(\int_0^1u^2du=\frac13\).
Replace \(x\) by \(a+b-x\) and add the two forms of the integral – many trigonometric integrals collapse to \(\int_a^b1\,dx\).
\(\int x\sin x\,dx\neq\int x\,dx\cdot\int\sin x\,dx\). Products need substitution or by parts.
\(\int x^{-1}dx=\log|x|+C\), but \(\int x^{-2}dx=-x^{-1}+C\) – the log rule is only for the power −1.
After substituting u = g(x) in a definite integral, either change the limits to u-values or substitute back before using the original limits.
In \(\int x e^x dx\), taking \(u=e^x\) makes the integral harder. ILATE says take \(u=x\) (algebraic before exponential).
\(\int\frac{x+1}{(x-1)(x-3)}dx\).
\(\frac{x+1}{(x-1)(x-3)}=\frac{-1}{x-1}+\frac{2}{x-3}\). Integral \(=-\log|x-1|+2\log|x-3|+C=\log\frac{(x-3)^2}{|x-1|}+C\).
\(\int_0^{\pi/2}\frac{dx}{1+\cot^5x}\).
Write the integrand as \(\frac{\sin^5x}{\sin^5x+\cos^5x}\). By P4 the same integral equals \(\int_0^{\pi/2}\frac{\cos^5x}{\cos^5x+\sin^5x}dx\). Adding the two forms gives \(2I=\int_0^{\pi/2}1\,dx=\frac\pi2\), so \(I=\frac\pi4\).
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