Integrals Class 12: notes and important questions

Integrals — revision notes

1. Standard integrals

• \(\int x^n dx = \dfrac{x^{n+1}}{n+1} + C\ (n \ne -1)\), \(\int\frac{dx}{x} = \log|x| + C\), \(\int e^x dx = e^x + C\), \(\int a^x dx = \dfrac{a^x}{\log a} + C\).
• \(\int\sin x = -\cos x\), \(\int\cos x = \sin x\), \(\int\sec^2x = \tan x\), \(\int\tan x = \log|\sec x|\), \(\int\sec x = \log|\sec x + \tan x|\), \(\int\csc x = \log|\csc x - \cot x|\).

2. Special forms

• \(\int\frac{dx}{x^2 + a^2} = \frac1a\tan^{-1}\frac xa\); \(\int\frac{dx}{x^2 - a^2} = \frac{1}{2a}\log\left|\frac{x - a}{x + a}\right|\); \(\int\frac{dx}{a^2 - x^2} = \frac{1}{2a}\log\left|\frac{a + x}{a - x}\right|\).
• \(\int\frac{dx}{\sqrt{a^2 - x^2}} = \sin^{-1}\frac xa\); \(\int\frac{dx}{\sqrt{x^2 \pm a^2}} = \log\left|x + \sqrt{x^2 \pm a^2}\right|\).
• \(\int\sqrt{a^2 - x^2}dx = \frac x2\sqrt{a^2 - x^2} + \frac{a^2}{2}\sin^{-1}\frac xa\); \(\int\sqrt{x^2 \pm a^2}dx = \frac x2\sqrt{x^2 \pm a^2} \pm \frac{a^2}{2}\log\left|x + \sqrt{x^2 \pm a^2}\right|\).
• Quadratic \(ax^2 + bx + c\) in the denominator (or under a root): complete the square. For \(\frac{px + q}{ax^2 + bx + c}\): write \(px + q = A\frac{d}{dx}(ax^2 + bx + c) + B\).

3. Techniques

• Substitution: look for a function and its derivative; change the limits in definite integrals.
• Partial fractions (proper fractions): \(\frac{A}{x-a} + \frac{B}{x-b}\); repeated factor \(\frac{A}{x-a} + \frac{B}{(x-a)^2}\); irreducible quadratic \(\frac{Bx + C}{x^2 + bx + c}\). Divide first if the fraction is improper.
• By parts: \(\int uv\,dx = u\int v\,dx - \int\left(u'\int v\,dx\right)dx\); choose \(u\) by ILATE. \(\int e^x[f(x) + f'(x)]dx = e^xf(x) + C\).

4. Definite integrals

If \(F' = f\) and \(f\) is continuous on \([a, b]\): \(\int_a^bf = F(b) - F(a)\). If \(A(x) = \int_a^xf(t)dt\), then \(A'(x) = f(x)\).

• \(\int_a^bf(x)dx = \int_a^bf(a + b - x)dx\); \(\int_0^af(x)dx = \int_0^af(a - x)dx\).
• \(\int_{-a}^af = 2\int_0^af\) if \(f\) is even, \(0\) if \(f\) is odd.
• \(\int_0^{2a}f = 2\int_0^af\) if \(f(2a - x) = f(x)\), \(0\) if \(f(2a - x) = -f(x)\).
• Split at points where a modulus or greatest-integer expression changes form.

Worked example 1

\(\displaystyle\int\frac{3x + 1}{x^2 + 2x + 5}dx\): \(3x + 1 = \frac32(2x + 2) - 2\), giving \(\frac32\log(x^2 + 2x + 5) - \tan^{-1}\frac{x + 1}{2} + C\).

Worked example 2

\(\displaystyle\int x^2e^{x}dx = x^2e^x - 2\int xe^x dx = e^x(x^2 - 2x + 2) + C\).

Worked example 3

\(\displaystyle I = \int_0^{\pi/2}\frac{\cos^4x}{\sin^4x + \cos^4x}dx\): using \(x \to \frac\pi2 - x\) and adding, \(2I = \frac\pi2\), so \(I = \frac\pi4\).

Common errors

• Forgetting \(+C\) in indefinite integrals, or keeping old limits after a substitution.
• \(\int\frac{dx}{2x + 3} = \frac12\log|2x + 3|\), not \(\log|2x + 3|\).
• Applying partial fractions to an improper fraction without dividing first.
• Using \(F(b) - F(a)\) when \(f\) is discontinuous in \([a, b]\).
• Sign slips with \(\int\sin x\,dx = -\cos x\).

Board tips

• For "property" questions write the property used — it carries a mark.
• Check an indefinite integral by differentiating your answer.
• Definite integrals as the limit of a sum and integrals of the type \(\int(px + q)\sqrt{ax^2 + bx + c}\,dx\) are no longer in the syllabus.