Important rules of integration for class 12th
Now wait a minute. Are you just posting random questions to keep a bunch of retired teachers occupied?
No sir. Its not so. Sorry for the disturbance.
Integration is an important concept in calculus, and there are several rules that can help you solve integration problems more efficiently. Here are some important rules of integration for Class 12th:
1. Power Rule:
- If you have an expression of the form x^n, where n is any real number except -1, then the integral of x^n with respect to x is (x^(n+1))/(n+1) + C, where C is the constant of integration.
2. Linearity Rule:
- If you have a sum or difference of multiple functions being integrated, you can integrate each function separately. For example, the integral of (f(x) + g(x))dx = ∫f(x)dx + ∫g(x)dx, and the integral of (f(x) - g(x))dx = ∫f(x)dx - ∫g(x)dx.
3. Constant Rule:
- If you have a constant multiplied by a function being integrated, you can bring the constant outside the integral. For example, if c is a constant, then the integral of c*f(x)dx = c*∫f(x)dx.
4. Integration by Substitution:
- If you have a composition of functions in the integrand, u-substitution can be used to simplify the integral. Let u be a function of x, and if the integrand can be written in the form f(u)*du, then you can substitute u for the inner function and replace dx with du/dx*dx in the integral. After making the substitution, integrate with respect to u and then replace u with the original function in terms of x.
5. Integration by Parts:
- The integration by parts formula allows you to find the integral of the product of two functions. If you have the integral of u(x)v'(x)dx, you can use the formula ∫u(x)v'(x)dx = u(x)v(x) - ∫v(x)u'(x)dx, where u(x) and v(x) are functions, and u'(x) and v'(x) are their respective derivatives.
It is important to practice using these rules and understand when to apply each one depending on the specific problem. Remember that integration is not always straightforward and may require creativity and trial-and-error.