Integrals Class 12 Notes- Chapter 7

Integrals

Integration is referred to the process which is inverse to that of differentiation. In the integration, we will be responsible for finding the function whose differential is provided to us. Integrals are the functions which satisfy a given differential equation.

What are Indefinite Integrals?

If integration is inverse of differentiation, then \(\frac{\partial }{\partial x}F(x)=f(x).\; Then \int f(x)dx=F(x) + C\) and these are referred as indefinite or general integrals.are all these differ by a constant term. If seen from a geometrical perspective, an indefinite integral is a compilation of curves which are obtained by translation of one of the curves parallel to itself downward and upward along with y-axis.

Properties of Indefinite Integrals

Integrals follow certain properties as follows:

  • \(\int [f(x)+g(x)dx]=\int f(x)dx+\int g(x)dx\)
  • For any given real number r, \(\int r f(x)dx]=r\int f(x)dx\)
  • If f1, f2, f3 … are the functions and r1,r2,r3… are the designated real numbers. Then,
\(\int [r_{1}f_{1}(x)+r_{2}f_{2}(x)+…+r_{n}f_{n}(x)]dx=r_{1}\int f_{1}(x)dx+r_{2}\int f_{2}(x)dx+…+r_{n}\int f_{n}(x)dx\)

Integration using Partial Fractions

The rational function property can be seen in fractions and is followed by the ratio of two polynomials such as R(x)/S(x) where x and S(x) \(\neq\) 0. If R(x) degree is higher than S(x), then we should divide R(x) by S(x) so that R(x)/S(x)=T(x)+ \(R_{1}(x)/S(x)\) , where T(x) is a degree x polynomial and the degree of \(R_{1}(x)\) is less than that of S(x). T(x) can be easily integrated.

Integration using Substitution

In this method, we change the variable of another variable in order to reduce the integral into fundamental integrals. We can get some standard integrals as the following.

  • \(\int tan x dx=log\left | secx \right |+C\)
  • \(\int cot x dx=log\left | sinx \right |+C\)
  • \(\int sec x dx=log\left | sec x + tan x\right |+C\)
  • \(\int cosec x dx=log\left | cosec x – cot x\right |+C\)

 

Important Questions:

Integrals Class 12 Notes Chapter 7

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