Final answer:
To express the integrand as a rational function, make the substitution u = sin x. Then solve for A and B in the partial fractions decomposition. Finally, substitute back u = sin x and integrate with respect to x.
Step-by-step explanation:
To evaluate the integral ∫(cos x)/(6 sin² x + 7 sin x) dx, we can make a substitution to express the integrand as a rational function. Let u = sin x, then du = cos x dx. Substituting these into the integral, we get: ∫(1/(6 u² + 7 u)) du. Now we have a rational function, which can be integrated using partial fractions.
Writing the rational function as 1/(6 u² + 7 u) = A/u + B/(6u + 7), we can solve for A and B by equating the numerators and simplifying. After finding the values of A and B, we can rewrite the rational function as (∫(A/u) du) + (∫(B/(6u + 7)) du).
Now we can evaluate these integrals using the natural logarithm function. The final step is to substitute back u = sin x and integrate with respect to x. This will give us the solution to the original integral.