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Math Analysis 1

Indefinite integrals: basic integrals and integration techniques

Antiderivatives, table of basic integrals, integration by substitution and by parts, rational functions with partial fractions, powers of sine and cosine.

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Integration is the inverse of differentiation: given f, we look for a function F whose derivative is f. While differentiating is mechanical, integrating requires recognising patterns and choosing the right method. This lesson collects the basic integrals and the three main methods: substitution, integration by parts, rational functions.

Antiderivatives and the indefinite integral

F is an antiderivative of f on an interval if F' = f. Two antiderivatives differ by a constant (a consequence of the mean value theorem), so the set of all antiderivatives is written \int f(x)\,dx = F(x) + C. Always check the result by differentiating it.

Basic integrals

\int x^\alpha\,dx = \frac{x^{\alpha+1}}{\alpha+1} + C (\alpha \ne -1)\int \frac{1}{x}\,dx = \ln|x| + C
\int e^x\,dx = e^x + C\int a^x\,dx = \frac{a^x}{\ln a} + C
\int \sin x\,dx = -\cos x + C\int \cos x\,dx = \sin x + C
\int \frac{1}{\cos^2 x}\,dx = \tan x + C\int \frac{1}{1+x^2}\,dx = \arctan x + C
\int \frac{1}{\sqrt{1-x^2}}\,dx = \arcsin x + C\int \frac{f'(x)}{f(x)}\,dx = \ln|f(x)| + C

The last entry is a case of the general rule \int g'(f(x))\,f'(x)\,dx = g(f(x)) + C: when a function appears together with its derivative, the integral is almost immediate. Examples: \int 2x\,e^{x^2}dx = e^{x^2} + C, \int \tan x\,dx = -\ln|\cos x| + C.

Integration by substitution

Set x = \varphi(t), so dx = \varphi'(t)\,dt, and substitute everything. At the end go back to the variable x.

    \[ \int \frac{1}{\sqrt{x}\,(1 + x)}\,dx \quad \xrightarrow{\ x = t^2,\ dx = 2t\,dt\ } \quad \int \frac{2t}{t(1 + t^2)}\,dt = 2\arctan t + C = 2\arctan\sqrt{x} + C. \]

Recurring substitutions: t = \sqrt{\dots} to remove a root, t = e^x for functions of e^x, t = \tan\frac{x}{2} for rational functions of sine and cosine.

Integration by parts

From the product rule we get

    \[ \int f'(x)\,g(x)\,dx = f(x)\,g(x) - \int f(x)\,g'(x)\,dx. \]

Choose as g the factor that becomes simpler when differentiated (polynomials, logarithms) and as f' the one that is easy to integrate.

  • \int x e^x\,dx = x e^x - \int e^x\,dx = (x - 1)e^x + C.
  • \int \ln x\,dx = \int 1\cdot\ln x\,dx = x\ln x - \int x\cdot\frac1x\,dx = x\ln x - x + C.
  • \int e^x\sin x\,dx: applying integration by parts twice gives back the original integral, and you solve for it as an equation: \frac{e^x(\sin x - \cos x)}{2} + C.

Rational functions

For \int \frac{P(x)}{Q(x)}dx: if the degree of P is greater than or equal to that of Q, divide first. Then split into partial fractions according to the roots of the denominator.

    \[ \frac{1}{x^2 - 1} = \frac{1}{(x-1)(x+1)} = \frac{1/2}{x - 1} - \frac{1/2}{x + 1} \quad\Rightarrow\quad \int \frac{dx}{x^2-1} = \frac12\ln\left|\frac{x-1}{x+1}\right| + C. \]

If the denominator has a quadratic factor with no real roots, such as x^2 + 2x + 5 = (x+1)^2 + 4, complete the square and you reach an arctangent: \int\frac{dx}{(x+1)^2 + 4} = \frac12\arctan\frac{x+1}{2} + C.

Trigonometric functions

Even powers of sine and cosine are reduced with \sin^2 x = \frac{1 - \cos 2x}{2} and \cos^2 x = \frac{1 + \cos 2x}{2}: \int \sin^2 x\,dx = \frac{x}{2} - \frac{\sin 2x}{4} + C. This is exactly the computation that gives the RMS value of a sinusoid, V_{\text{rms}} = \frac{V_m}{\sqrt 2}.

Not every elementary function has an elementary antiderivative: e^{-x^2}, \frac{\sin x}{x}, \frac{1}{\ln x} cannot be integrated with these methods. Their integrals define new functions (the error function of statistics, the sine integral of filter theory) and are computed numerically.

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