Properties of definite integrals
\(\int_0^af(x)dx=\int_0^af(a-x)dx\); \(\int_{-a}^af=2\int_0^af\) (even) or 0 (odd).
Integrals
Properties of Definite Integrals
Understand
| Property | Statement |
|---|---|
| P1 | \(\int_a^bf=-\int_b^af\), \(\int_a^af=0\) |
| P2 | \(\int_a^bf=\int_a^cf+\int_c^bf\) |
| P3 | \(\int_a^bf(x)\,dx=\int_a^bf(a+b-x)\,dx\) |
| P4 | \(\int_0^af(x)\,dx=\int_0^af(a-x)\,dx\) |
| P5 | \(\int_{-a}^af=2\int_0^af\) if \(f\) is even; \(0\) if \(f\) is odd |
| P6 | \(\int_0^{2a}f=2\int_0^af\) if \(f(2a-x)=f(x)\); \(0\) if \(f(2a-x)=-f(x)\) |
Classic use of P4: \(I=\int_0^{\pi/2}\frac{\cos^3x}{\sin^3x+\cos^3x}dx\). Replacing \(x\) by \(\frac\pi2-x\) gives \(I=\int_0^{\pi/2}\frac{\sin^3x}{\cos^3x+\sin^3x}dx\). Adding, \(2I=\int_0^{\pi/2}1\,dx=\frac\pi2\), so \(I=\frac\pi4\).
Modulus and piecewise functions: split the interval at the points where the expression changes sign (P2): \(\int_{-1}^2|x|\,dx=\int_{-1}^0(-x)\,dx+\int_0^2x\,dx=\frac12+2=\frac52\).
Key Concepts
- Odd function on [−a, a] ⇒ 0.
- P4 + add the two forms of I.
- Split at sign changes for |f(x)|.
Formula Bank
Key Points
King’s property
Replace \(x\) by \(a+b-x\) and add the two forms of the integral – many trigonometric integrals collapse to \(\int_a^b1\,dx\).
Solved Examples
Using P4
\(\int_0^{\pi/2}\frac{dx}{1+\cot^5x}\).
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 \(I=\frac\pi4\).
Practice & Topic Test
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Before integrating, look for symmetry (odd/even) or the a − x trick; split intervals for modulus functions.
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