Integration of the Function
To solve the integral
\[
\int \frac{\sin x + \cos x}{\sin^2 x + \cos^4 x} \, dx,
\]
we will simplify the integrand before integrating.

Step 1: Rewrite the Denominator
- We start with the expression in the denominator:
\[
\sin^2 x + \cos^4 x = \sin^2 x + (\cos^2 x)^2.
\]
- This can be viewed as a sum of squares.

Step 2: Use a Trigonometric Identity
- Recall the identity:
\[
\sin^2 x + \cos^2 x = 1.
\]
- This can help us in expressing \(\sin^2 x\) in terms of \(\cos^2 x\).

Step 3: Substitution Technique
- Let \( u = \sin x + \cos x \). Then, the derivative \( du = (\cos x - \sin x) \, dx \).
- Rearranging gives \( dx = \frac{du}{\cos x - \sin x} \).

Step 4: Substitute in the Integral
- Substitute \( u \) and transform the integral accordingly. The integral becomes:
\[
\int \frac{u}{\sin^2 x + \cos^4 x} \cdot \frac{du}{\cos x - \sin x}.
\]
- This can be simplified further depending on the specific values of \( u \).

Final Step: Integrate
- After simplification and substitution, integrate with respect to \( u \).
- This integral can be solved using standard techniques or recognized forms.
The solution will yield a function plus a constant of integration \( C \). Always remember to check your work by differentiating the result to ensure it matches the original integrand.