(i) The process of differentiation and integration are inverses of each other.
i.e. and
(ii) Two indefinite integrals with the same derivative lead to the same family of curves, and so they are equivalent.
(iii) The integral of the sum of two functions is equal to the sum of integrals of the given functions,
i.e.
(iv) For any real value of p,
(v) For a finite number of functions f_{1}, f_{2 }…. f_{n} and the real numbers p_{1}, p_{2}…p_{n},
∫[p_{1}f_{1}(x) + p_{2}f_{2}(x)….+p_{n}f_{n}(x) ]dx = p_{1}∫f_{1}(x)dx + p_{2}∫f_{2}(x)dx + ….. + p_{n}∫f_{n}(x)dx
1. Integration by substitution: In this method the integral ∫f(x)dx is expressed in terms of another integral where some other variables say t is the independent variable; x and t being connected by some suitable relation x=g(t).
It leads to the result ∫f(x)dx=∫f(g(t)). g'(t) dt
2. Integration by parts: This method is used to integrate the product of two functions. If f(x) and g(x) are two integrable functions, then
i.e. The integral of (product of two functions) = first function * integral of the second  integral of (derivative of first function * integral of the second function)
In order to select the first function, the following order is followed:
Inverse → Logarithmic → Algebraic → Trigonometric → Exponential
3. Integration by a partial fraction: If the integral is in the form of an algebraic fraction that cannot be integrated then the fraction needs to be decomposed into partial fractions.
Rules for expressing in partial fraction:
Integral Calculus is mainly used for the following two purposes:
We had differentiation formulas, we have integral formulas as well. Let us go ahead and look at some of the integral calculus formulas.
The important application of integral calculus are as follow:
Example: Find the integral for the for the following function.
(i) f(x) = √x
(ii) f(x) =cos^{2}x
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1. What is the difference between definite and indefinite integrals in integral calculus? 
2. What are some common methods of integration used in integral calculus? 
3. How is integral calculus used in realworld applications, particularly in commerce? 
4. Can you provide some examples of integral calculus applications in commerce? 
5. What are some key integral calculus formulas that are frequently used in commerce applications? 
209 videos443 docs143 tests


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