Final answer:
To solve the given equation, we need to apply the correct trigonometric substitution and simplify the integral. The integral can be rewritten using the identity tan^2(x) = sec^2(x) - 1 and then applying the substitution u = tan(x) and du = sec^2(x) dx. Finally, we determine the limits of integration by substituting the original limits in terms of x into the equation u = tan(x).
Step-by-step explanation:
To solve the equation:
I = ∫0π/4 44(x) tan2(x) dx = ∫ab up(1+u2)ᵢ du
we need to apply the correct trigonometric substitution. Let's start by simplifying the integral:
I = ∫0π/4 44(x) tan3(x) dx
Using the identity tan2(x) = sec2(x) - 1, we can rewrite the integral as:
I = ∫0π/4 44(x) (sec2(x) - 1) tan(x) dx
Now, we can let u = tan(x), and du = sec2(x) dx:
I = ∫0π/4 44(u) (u2 - 1) du
The next step is to determine the limits of integration a and b, which depend on the original limits of integration in terms of x. Since the original limits are from 0 to π/4, we can substitute x = 0 and x = π/4 into the equation u = tan(x) to find the corresponding values of u. The limits of integration become a = tan(0) and b = tan(π/4):
a = tan(0) = 0
b = tan(π/4) = 1
Now, we can rewrite the integral again:
I = ∫ab up(1+u2)ᵢ du