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Integration

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What is Integration Edit

Integration can be defined as a process of bringing small quantities together and finding the overall result. For instance, it is possible to find the area of a whole body without knowing its exact dimensions. All we need to know are the dimensions of the single particle of that body. By integrating over a certain boundary value, the resultant outcome will be for instance the area of the whole body.

Integrations have important applications in physics, chemistry and in engineering giving us accurate results using less mathematical operations in a shorter period of time. Imagine how many complicated steps are involved just to compute the area of each individual particle of a given body and then add them all up. With integration techniques the results are obtained quicker just by knowing the limits of the area to be integrated. Accurate results can be obtained if integration techniques are applied properly.


By simple substitution Edit

This is one of the easiest ways to solve an integral problem. The characteristic of this technique is that it can be used by parts, as partial fractions and as trigonometric manipulations. Before starting to solve the problem just by applying simple integration techniques, one should first detect whether the problem can be solved by substitutions. To do this, the numerator and denominator of the integrant have to be carefully analyzed. For instance, the following problem demonstrates how substitution is being applied.

∫(cosx)dx/(1+sin2x)


The problem just shown looks similar to the Trigonometric Manipulation discussed in the next subheading. It can be easily solved by substituting 1+sin2x with u. The fact that 1 is a constant and its derivative is 0, all is left for u’will be the sin2x term. Doing this, the derivative of sin2x will be cos2x which corresponds to the numerator. So, this problem is a good candidate for the substitution.


Trigonometric manipulation Edit

This is another technique to solve integral problems involving trigonometric functions. Trigonometric integrals cannot be always solved using substitution. For instance, consider the following problem:

∫(cos3x)(sin4x)dx


This problem is a trigonometric manipulation problem. While solving such types of problems, it is good idea to transform odd power term to even power terms leaving one term at the end of the integral. If we apply this to above example, it will become

∫(cos2x)(sin4x)(cosx)dx


The above problem now becomes simple, because we know the relation between sin x and cosx, which is

sin2x+cos2x=1


Integration by parts Edit

These integrals are easy to detect because an integrant is a product of two functions, such as, trigonometric, exponential, algebraic, logarithmic, and so on. Substitution does not always work to solve these types of problems. The general formula to solve this type of integrations is:

∫udv = uv - ∫vdu


The only trick in integration by parts is to choose appropriate values for u and for vdv. Choosing the wrong u-substitution from the product term of the function, will most likely make the computation impossible to solve.

Hint: If there is product of two terms in a function and one of the terms is logarithmic, one should use the logarithmic term for the u-substitution instead of the v-substitution

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