## Saturday, September 22, 2012

### Trigonometric Integrals

Trigonometry is a fundamental concept of mathematics. It is used in calculus functions and vectors. In this topic we have to use trigonometry as integral function. That means how to integrate trigonometric functions. For this we also have to know what is integration?  Integration means to calculate area of a given curve, and the curve is a closed curve made by x axis and y axis.

Trigonometric integrals mean integration of trigonometric functions. As we know these trigonometric functions are basic formulas for solving trigonometric integral. To more simplify this term, let’s take an example like sin2X. This is a trigonometric function. And we integrate this function for this first we have to expand this term by using formula of trigonometry. After expanding we carry out the constant term then by using product rule of integral, we can integrate this trigonometric function.

Above example is simple it has only one trigonometric function but trigonometric function may be combine with other function also. It can be algebraic function with trigonometry, logarithmic function with trigonometry and exponential function with trigonometry. These are also called integrals of trigonometric functions. To solve this type of problem either we can use integration by substitution method or integration by parts method.

Inverse trigonometric integrals such as sin^-1X and cos ^-1X etc. now to integrate this type of functions we have to use basics of calculus. We need  to take this function equal to any constant like Y. means we have to write Y= sin ^-1X. now we transfer sin function to other site the we get. X=sin Y. Now we can simply integrate this term.

Trigonometric substitution integrals, here we also integrate trigonometric functions and calculus functions, but procedure is different. To integrate this type of function we have to substitute and equal trigonometric term in place of other trigonometric term. The first from of integrals is integration of [f’(x)/f(x)] dx=logf(x) . In this form integral of a function whose numerator is the exact derivative of its denominator and equal to the logarithmic of its denominator? The second form is, in the integrand consist of the product of a constant power of a function f(x) and the derivative of f(x), to obtain the integral we increase the index by unity and then divide by increase index. This procedure is known as power formula. Lets take an example suppose we have to integrate (4x^3/1+x^4) dx= ln (1+x^4). By using this method we substitute 1+x^4 = any constant term like (t), and after that we integrate this function.