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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