Chain rule can be defined as:
Substitution rule is exact opposite of chain rule for differentiation in Integration.
Suppose that we have an integral such as:
There are no anti Differentiation Formulas but from our knowledge of differentiation, specifically the chain rule, we know that 4x3 is the derivative of the function within the Square root, x4 + 7. We must also account for the chain rule when we are performing integration. To do this, we use the substitution rule. The Substitution Rule states: if u = g(x) is a differentiable function and f is continuous on The Range of g, then,
So integration of 4x^3*
We have to follow some of the simple rules by looking at Derivatives of the Functions: example the antiderivative of x^k will be x^k+1/ka+1.A function F is an antiderivative of f on an interval I, if d/dxF(x) = f(x). This is a strong indication that that the processes of integration and differentiation are interconnected. We know Integral of Cosx is defined as Sinx +c. So antiderivative of cos2x will be Sin2x/2 + c following chain rule and substitution rule.
Some of the more examples of Integrals of Trigonometric Functions are:
1> ∫Sinx dx=-Cosx +c
2> ∫Sec^2 x dx=Tanx+c
3> ∫Cosec^2 x dx=-Cotx+c
4> ∫Secx tanx dx=Sec x +c
5> ∫Cosecx Cot x dx=-Cosec x +c
Subsequent to finding an indefinite integral, constantly make sure if your answer is accurate. Since integration and differentiation are opposite processes, you can plainly differentiate the function that consequences from integration, and see if it is equal to the Integrand.