 Hello and welcome to the session. In this session we will discuss properties of indefinite integrals. The first property is, as we know that process of differentiation and integration are inverses of each other. So we get that d by dx of integral fx dx is equal to fx and integral f dash x dx is equal to fx plus c. And this c is any arbitrary constant. The next property is two indefinite integrals with the same derivative lead to the same family of curves and so they are equivalent. Property is integral fx plus gx dx is equal to integral fx dx plus integral gx dx. Then we have for any real number k integral k fx dx is equal to k into integral fx dx. Then next we have the properties 3 and 4 can be generalized to finite number of functions f1 f2 and so on up to fn and real numbers k1 k2 and so on up to kn. Giving integral k1 f1 x plus k2 f2 x plus and so on up to kn fn x dx is equal to k1 integral f1 x dx plus k2 integral f2 x dx plus and so on up to kn. Let's try and find out the value of i equal to integral 2x square plus e to the power x dx. Here we have two functions in the integrand 2x square is one function and e to the power x is the second function. Now using this third property according to which we have integral fx plus gx dx is equal to integral fx dx plus integral gx dx. We get that i is equal to integral 2x square dx plus integral e to the power x dx. Now for this integral we will use this fourth property which says that for any real number k integral k fx dx is equal to k into integral fx dx. So this becomes equal to 2 integral x square dx plus integral ex dx. Now this is equal to 2x cube upon 3 plus e to the power x plus c. That is we have i is equal to 2 upon 3x cube plus e to the power x plus c where the c is the constant of integration. Thus this is how we use the properties of indefinite integrals to find the value of the integrals. Next we shall discuss the comparison between differentiation and integration. First of all we have that both are operations on functions. So this is a kind of similarity. Then next we have both satisfy the property of linearity according to which we have d by dx of k1 f1x plus k2 f2x is equal to k1 d by dx of f1x plus k2 d by dx of f2x. And integral k1 f1x plus k2 f2x dx is equal to k1 integral f1x dx plus k2 integral f2x dx. Here these k1 k2 are the constants and we know that all functions are not differentiable and also all functions are not integrable. Next we have derivative of a function when it exists is a unique function and integral of a function is unique to an additive constant. That is the two integrals of a function differ by a constant. Next is when a polynomial function p is differentiated the result is a polynomial whose degree is 1 less than the degree of p. When a polynomial function is integrated the result is a polynomial whose degree is 1 more than that of p. Next we have we can speak of the derivative at a point but we never speak of the integral at a point. Then integral of a function is always over an interval on which the integral is defined. Then the derivative of a function has a geometrical meaning namely slope of the tangent to the corresponding curve at a point and the indefinite integral of a function represents a family of curves. This completes the session. Hopefully you have understood the properties of indefinite integral and the comparison between differentiation and integration.