খসড়া:গাজা গণহত্যা

Techniques of integration

Finding antiderivatives of elementary functions is often considerably harder than finding their derivatives (indeed, there is no pre-defined method for computing indefinite integrals). For some elementary functions, it is impossible to find an antiderivative in terms of other elementary functions. To learn more, see elementary functions and nonelementary integral.

There exist many properties and techniques for finding antiderivatives. These include, among others:

• The linearity of integration (which breaks complicated integrals into simpler ones)
• Integration by substitution, often combined with trigonometric identities or the natural logarithm
• The inverse chain rule method (a special case of integration by substitution)
• Integration by parts (to integrate products of functions)
• Inverse function integration (a formula that expresses the antiderivative of the inverse টেমপ্লেট:Math of an invertible and continuous function টেমপ্লেট:Mvar, in terms of the antiderivative of টেমপ্লেট:Mvar and of টেমপ্লেট:Math).
• The method of partial fractions in integration (which allows us to integrate all rational functions—fractions of two polynomials)
• The Risch algorithm
• Additional techniques for multiple integrations (see for instance double integrals, polar coordinates, the Jacobian and the Stokes' theorem)
• Numerical integration (a technique for approximating a definite integral when no elementary antiderivative exists, as in the case of টেমপ্লেট:Math)
• Algebraic manipulation of integrand (so that other integration techniques, such as integration by substitution, may be used)
• Cauchy formula for repeated integration (to calculate the টেমপ্লেট:Math-times antiderivative of a function) $$\underset{x_0}{\overset{x}{\int }}\underset{x_0}{\overset{x_1}{\int }}\cdots \underset{x_0}{\overset{x_{n-1}}{\int }}f(x_n)\mathrm{d}{x}_n\cdots \mathrm{d}{x}_2\mathrm{d}{x}_1=\underset{x_0}{\overset{x}{\int }}f(t)\frac{(x-t{)}^{n-1}}{(n-1)!}\mathrm{d}t.$$

Computer algebra systems can be used to automate some or all of the work involved in the symbolic techniques above, which is particularly useful when the algebraic manipulations involved are very complex or lengthy. Integrals which have already been derived can be looked up in a table of integrals.

Of non-continuous functions

Non-continuous functions can have antiderivatives. While there are still open questions in this area, it is known that:

• Some highly pathological functions with large sets of discontinuities may nevertheless have antiderivatives.
• In some cases, the antiderivatives of such pathological functions may be found by Riemann integration, while in other cases these functions are not Riemann integrable.

Assuming that the domains of the functions are open intervals:

• A necessary, but not sufficient, condition for a function টেমপ্লেট:Math to have an antiderivative is that টেমপ্লেট:Math have the intermediate value property. That is, if টেমপ্লেট:Math is a subinterval of the domain of টেমপ্লেট:Math and টেমপ্লেট:Math is any real number between টেমপ্লেট:Math and টেমপ্লেট:Math, then there exists a টেমপ্লেট:Mvar between টেমপ্লেট:Mvar and টেমপ্লেট:Mvar such that টেমপ্লেট:Math. This is a consequence of Darboux's theorem.
• The set of discontinuities of টেমপ্লেট:Math must be a meagre set. This set must also be an F-sigma set (since the set of discontinuities of any function must be of this type). Moreover, for any meagre F-sigma set, one can construct some function টেমপ্লেট:Math having an antiderivative, which has the given set as its set of discontinuities.
• If টেমপ্লেট:Math has an antiderivative, is bounded on closed finite subintervals of the domain and has a set of discontinuities of Lebesgue measure 0, then an antiderivative may be found by integration in the sense of Lebesgue. In fact, using more powerful integrals like the Henstock–Kurzweil integral, every function for which an antiderivative exists is integrable, and its general integral coincides with its antiderivative.
• If টেমপ্লেট:Math has an antiderivative টেমপ্লেট:Math on a closed interval $[a,b]$, then for any choice of partition $a=x_0<x_1<x_2<\dots <x_n=b,$ if one chooses sample points $x_i^{*}\in [x_{i-1},x_i]$ as specified by the mean value theorem, then the corresponding Riemann sum telescopes to the value $F(b)-F(a)$. $$\begin{array}{cc}\sum _{i=1}^{n}f(x_i^{*})(x_i-x_{i-1})& =\sum _{i=1}^n[F(x_i)-F(x_{i-1})]\\ & =F(x_n)-F(x_0)=F(b)-F(a)\end{array}$$ However, if টেমপ্লেট:Math is unbounded, or if টেমপ্লেট:Math is bounded but the set of discontinuities of টেমপ্লেট:Math has positive Lebesgue measure, a different choice of sample points $x_i^{*}$ may give a significantly different value for the Riemann sum, no matter how fine the partition. See Example 4 below.