Why does the indefinite integral require +c on the end of its solutions? Why is

Question

1. Why does the indefinite integral require +c on the end of its solutions? Why is the + c not needed for a definite integral?
2. What applications can you find either in your program of study or elsewhere that require integration to reach a solution?
3. Why is (x+5)3 a correct anitiderivative of 3(x+5)2, whereas (2x+5)3 is not a correct antiderivative of 3(2x+5)2?
4. What is the power rule for integration? How do we use it?
5. How do we recognize an integral whose solution will be a logarithmic form? How do we solve such an integral?
6. How do we use the formula for the integral of e^x to integrate functions of this form?
7. How do we use the basic trig integration formulas to integrate more complex trig functions?
8. What role does substitution play in integration with logarithmic, exponential, and trig forms?
9. Why is Calculus, and in particular, Integral Calculus, the foundation for the study of physics? In other words, why did Sir Isaac Newton develop Calculus?

Explanation / Answer

1. +c is an unknown constant which arises since we do not know the limits of the integration. In definite integral, we know these limits and hence +c vanishes.

2.Used to find areas, volumes,work done by variable force,force by a liquid pressure etc.

3. Because we cannot ignore the constant multiplied by the variable whereas a constant in addition with it is treated differently.In derivatives,you need to multiply the value obtained by the coefficient of the variable.

4.In power rule, integration of x^n gives x^n+1/n+1

5.Whenever there is x in its first power in the denominator, it will lead to a logarithmic solution. Integration of 1/x gives loge x.

6. Both integration and differentiation of e^x gives e^x.In the product form of integration, it can be taken as the u term.

7.The best way is to break a function of trig int he form of simpler forms using the basic trig conversions.Also, one needs to remember basic trig and inverse trig functions.

8.Substitution is the most important concept in integration. A form in terms of a variable x can be made easier by substituting x with say, sin x.

Any tem in the denominator can be replaced by a single variable y to find its log.

Similarly for exponential functions.

9.This is clear from the applications of integration.Wherever we do not have a simple formula to solve over a given interval, we need to divide it into different smaller intervals and then sum them over that interval.In physics, whenever we have to find out micro particle rules, nuclear physics and universe physics, there are constantly varying forces and we need integartion.

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