Integration with substitution
The integral of f'(g(5x))g'(5x)dx
f(g(5x))/5
To solve the integral ∫f'(g(5x))g'(5x)dx using the substitution method, we need to perform a change of variables. Let's assume u = g(5x).
Now, we will differentiate both sides of the equation with respect to x. Since u is a function of x, we can use the chain rule:
du/dx = d(g(5x))/dx
To simplify this further, let's differentiate g(5x) with respect to x by applying the chain rule:
du/dx = (dg(5x)/d(5x)) * (d(5x)/dx)
du/dx = g'(5x) * 5
Now, we can see that du/dx = 5 * g'(5x). Rearranging this equation, we get:
g'(5x) = (1/5) * du/dx
Next, we need to differentiate f'(g(5x)) with respect to x. Again, applying the chain rule:
d(f'(g(5x)))/dx = (d(f'(g(5x)))/d(g(5x))) * (d(g(5x))/dx)
The term d(g(5x))/dx is 5 * g'(5x), as we derived earlier. So, substituting this value, we have:
d(f'(g(5x)))/dx = (d(f'(g(5x)))/d(g(5x))) * (5 * g'(5x))
= 5 * f''(g(5x)) * g'(5x)
Now, we can rewrite the integral as:
∫f'(g(5x))g'(5x)dx = ∫f''(g(5x)) * (5 * g'(5x)) dx
= ∫(5 * f''(g(5x)) * g'(5x)) dx
Since u = g(5x), the expression inside the integral can be rewritten as:
5 * f''(u) du
Substituting back the variable u, the integral becomes:
∫5 * f''(u) du
This integral can be easily evaluated with respect to u, resulting in:
5 * ∫f''(u) du
Now, you can apply the integration rules or techniques to integrate f''(u) with respect to u, and the final answer will be 5 times the result of that integration.