Calculus is a branch of mathematics that studies the change of one quantity in relation to another. There are two processes in calculus -- differentiation and integration. These processes are the opposite of each other and hence the result of integration produces an antiderivative. There are several ways to integrate a function to produce the antiderivative, and this depends upon the particular function.
Antiderivative of Simple Functions
Antiderivatives are found by integrating a function. If the function in question is simple, it should be found in an antiderivative table. To find the anti-derivative of a particular function, find the function on the left-hand side of the table and find the corresponding antiderivative in the right-hand side of the table. For example, if the antiderivative of cos(x) is required, the table shows that the anti-derivative is sin(x) + c.
Antiderivative of Products
Sometimes a function will be the product of two simpler functions. In this case an antiderivative table will not give a result. Instead, some simple mathematics will need to be carried out. The method to find the antiderivative of products is known as integration by parts. To do integration by parts, split the source equation into two products. For example, if the source function is y=x.cos(x), split into u=x and dv/dx=cos(x). The next step is to use an anti-derivative table to find the derivative of u and the antiderivative of dv/dx:
u=x, therefore the derivative du/dx=1
dv/dx=cos(x), therefore the antiderivative v=sin(x)
The following equation is then used to evaluate the antiderivative:
Integral(u.dv/dx dx) = u.v - Integral(v.du/dx dx)
Following the example:
Integral(x.cos(x))= x.sin(x) - Integral(sin(x)) =x.sin(x) + cos(x) + c
Antiderivative by Substitution
There are some occasions where a difficult antiderivative can be found by making an appropriate substitution. The best way to learn this technique is by example. If the antiderivative of y=(x+4)^5 is required, the use of an antiderivative table will not be of direct use. Instead it is best to make the substitution u=x+4. The problem has then been reduced to y=u^5, which can be looked up on an antiderivative table. The integral of y=u^5 is equal to (u^6)/6 du + c. To obtain du, u=x+4 is differentiated, which leads to du/dx=1. Rearranging yields dx=du, so in this case they are equal. Finally, the antiderivative is converted back to the original variable x: (u^6)/6 + c = ((x+4)^6)/6 + c.
Antiderivative by Trigonometric Identities
The basic trigonometric functions are sine, cosine and tangent formulae. An identity is a formula that allows a trigonometric function to be written in a different form. Sometimes the antiderivative of a complicated trigonometric function is required, and in this case an identity can be used to simplify the equation. For example, if the antiderivative of sin^2(x) is required, the "double angle" formula can be used to rewrite the equation: Integral(sin^2(x)) =Integral(0.5(1-cos(2x))). The right-hand side of the equation can be evaluated using a standard anti-derivative table. Following the example: Integral(0.5(1-cos(2x)))=0.5(x-0.5sin(2x)).
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