The first step in solving any integral is to identify the appropriate trigonometric identities that can be used to simplify the expression. In this case, we can use the identity tan^2x + 1 = sec^2x to rewrite the integral as I(sec^2x tanx/cos^4x, x).
After simplifying the expression, we can use the substitution method to solve the integral. Let u = tanx, which means du = sec^2x dx. Substituting these values into the integral, we get I(u^3/cos^4x, x) = I(u^3, u).
The key to solving this integral using substitution is to choose a substitution that will eliminate all instances of the variable in the integral. In this case, we chose u = tanx, which eliminated the cos^4x term in the denominator.
Integrating u^3 is a simple matter of applying the power rule of integration. This means that I(u^3, u) = (u^4)/4 + C. Remember to substitute back in the original variable x to get the final solution.
Yes, there is one final step to complete the solution. Now that we have the indefinite integral (u^4)/4 + C, we need to substitute back in the original variable x to get the final solution. This means that the final solution is (tan^4x)/4 + C.