 Hello and welcome to the session. In this session, we will discuss integration as an inverse process of differentiation. Integration is basically the inverse process of differentiation. In the differential calculus, we are given a function and we have to find the derivative or differential of this function. But in integral calculus, we are to find a function whose differential is given. Suppose we have d by dx of fx is given to be equal to fx, then we write integral of fx dx is equal to fx plus c. These integrals are called indefinite integrals and the c is the constant of integration. And all these integrals differ by a constant. We already know the formulae for the derivatives of many important functions. From these formulae, we can write down immediately the corresponding formulae for the integrals of these functions. Like we know that d by dx of x raised to the power n plus 1 over n plus 1 is equal to x raised to the power n, then this gives us that integral of x raised to the power n dx is equal to x raised to the power n plus 1 over n plus 1 that is this function plus the constant of integration c and this n is not equal to minus 1. In the same way from the formulae for the derivatives, we find the formulae for the integrals. Like we have integral of cos x dx is equal to sin x plus c, then integral of sin x dx is equal to minus cos x plus c. Next is integral of sin x square x dx is equal to tan x plus c, then the next is integral of cos x square x dx is equal to minus cos x plus c, then integral of sin x tan x dx is equal to sin x plus c. Next is integral of cos x cortex dx is equal to minus cos x plus c, next one is integral of dx upon square root 1 minus x square is equal to sin inverse x plus c, also integral dx upon square root 1 minus x square is equal to minus cos inverse x plus c, then we have integral dx upon 1 plus x square is equal to tan inverse x plus c, then integral dx upon 1 plus x square is equal to minus cos inverse x plus c, then integral e raise to the power x dx is equal to e raise to the power x plus c, next one is integral a to the power x dx is equal to a to the power x upon log a plus c, next one is integral dx upon x multiplied by square root x square minus 1 is equal to sin inverse x plus c, next one is integral dx upon x multiplied by square root x square minus 1 is equal to minus cos inverse x plus c, then integral 1 upon x dx is equal to log modulus x plus c, we need to find the integral i equal to integral of e to the power 4x dx, for this we will use the standard formula for integral e to the power x dx is equal to e to the power x plus c, as you can see that the coefficient of x here is 4, so we will divide the value of this integral by 4, this becomes equal to 1 upon 4 e to the power 4x plus c, this is the value for i and the c is the constant of integration, so in this way we can use the above 16 formulae to find the value of the integrals, now let's discuss the geometrical interpretation of indefinite integral, from geometric point of view an indefinite integral is collection of family of curves each of which is obtained by translating one of the curves parallel to itself upwards or downwards along the y axis, let fx be equal to 2x then integral fx dx is equal to x square plus c, now for different values of c we get different integrals but these integrals are similar geometrically, so we have y equal to x square plus c where c is the constant, this represents a family of integrals by assigning different values to c we get different members of the family and these together constitute the indefinite integral, in this case each integral represents the parabola with its axis along y axis, now for c equal to 0 in y equal to x square plus c we get y is equal to x square, as you can see that y equal to x square is a parabola with its vertex on the origin, for c equal to 1 we have y equal to x square plus 1, this is the curve y equal to x square plus 1 as you can see this is the parabola which is obtained by shifting the parabola y equal to x square 1 unit along y axis in positive direction, next we have for c equal to minus 1 we have y equal to x square minus 1, this is the curve for y equal to x square minus 1, as you can see this is also a parabola which is obtained from the parabola y equal to x square by shifting this parabola 1 unit along y axis in the negative direction, thus we see that for each positive value of c each parabola of the family has its vertex on the positive side of the y axis and for negative values of c each has its vertex along the negative side of the y axis, now let us consider the intersection of these parabolas by a line x equal to a, so this line x equal to a intersects these parabolas at points p naught p 1 p 2, then dy upon dx at these points p naught p 1 p 2 would be equal to 2a, this indicates that the tangents to the curves at these points are parallel, as you can see that the tangents at these points are parallel, thus we have integral 2x dx equal to x square plus c equal to say fc implies that the tangents to all the curves y equal to fcx where c belongs to r at the points of intersection of the curves by the line x equal to a where a belongs to r at parallel, thus we have by giving different values to c we can obtain different numbers of the family of curves. This completes the session, hope you understood the different formulae of integration and how do we get them from differentiation and the geometrical interpretation of indefinite integral.