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Inverse Chain Rule Method

We solve the mathematicsintegral by using inverse chain rule method. Inverse chain rule method is very useful to integrate the function if given function in this form integral of [f(x)]^n f’(x). In order to integration by using inverse chain rule method we should be noticed that integrand consist of the product of derivative f’(x) of f(x) and should be a power of a function f(x).Then we find integral by increasing exponent unity and divide by increased index. Let us see the formula of integration by using inverse chain rule: - integral of [f(x)]^n * f’(x) = [f(x)]^n+1 / (n + 1), here n is not equal to -1.

Let us see some examples of integration by using inverse chain rule method h(x) = (4x^4 + 2x^3 + 5x^2 + x + 1)^8 * (16x^3 + 6x^2 + 10x + 1). Solution: - Given function h(x) = (4x^4 + 2x^3 + 5x^2 + x + 1)^8 * (16x^3 + 6x^2 + 10x + 1). Here we let f(x) = 4x^4 + 2x^3 + 5x^2 + x + 1 then f’(x) = 16x^3 + 6x^2 + 10x + 1. We see that given integral in this form [f(x)]^n * f’(x). So that integration of h(x) = (4x^4 + 2x^3 + 5x^2 + x + 1)^8 * (16x^3 + 6x^2 + 10x + 1), using formula integral of [f(x)]^n * f’(x) = [f(x)]^n+1 / (n + 1), let us take integration and gate, Integral of (4x^4 + 2x^3 + 5x^2 + x + 1)^8 * (16x^3 + 6x^2 + 10x + 1) = (4x^4 + 2x^3 + 5x^2 + x + 1)^9 / 9 + C, here C is any constant..

By using integration inverse chain rule, we can integrate difficult integration easily. Let we see another example: -find integration of sin^5 x * cos x by using inverse chain rule method.Solution: - Given integral function sin^5 x * cos x, Here we take f(x) = sinx then f’(x) = cosx. Now we can write in this form of f(x) ^n f’(x). So we use fomula integration by inverse chain rule ` integral of [f(x)]^n * f’(x) = [f(x)]^n+1 / (n + 1). Now bu using this formula integral function sin^5 x * cos x = sin^6 x / 6 + C, here C is any constant. Let we see another example: - Find the integration by using chain rule method if h(x) = (3x + 1) / (x^3 + x + 4). Solution: - Given function h(x) = (3x^2 + 1) / (x^3 + x + 4), Here we let f(x) = x^3 + x + 4, now f’(x) = 3x^2 + 1, So we use formula and gate Integral of (3x + 1) / (x^3 + x + 4) = log (x^3 + x + 4) + C.

This is how the inverse chain rule can used. The illustrations stated above shows the steps of finding the integration of any function byusing the chain rule method.

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