Final answer:
To evaluate the indefinite integral ∫x cos x/sin dx, we can rewrite the integrand and split it into two separate integrals. The integral of x dx is (1/2)x² + C, and the integral of cos x/sin² x dx can be evaluated using substitution or trigonometric identities. The final result is (1/2)x² - (1/sin x) + C + C'.
Step-by-step explanation:
To evaluate the indefinite integral ∫x cos x/sin dx, we can start by rewriting the integrand using trigonometric identities. Using the identity cos x = sin π/2 - x, the integrand becomes x sin (π/2 - x)/sin x dx. We can then use the identity sin (π/2 - x) = cos x/sin x to simplify further. The integral now becomes ∫x (cos x/sin x)/sin x dx.
Next, we can split the integral into two separate integrals: ∫x dx and ∫(cos x/sin² x) dx. The integral of x dx is (1/2)x² + C, where C is the constant of integration. The second integral can be evaluated using substitution or trigonometric identities.
By applying the substitution u = sin x, du = cos x dx, the integral ∫(cos x/sin² x) dx can be rewritten as ∫(1/u²) du. Integrating this expression gives -1/u + C, where C is the constant of integration.
Therefore, the indefinite integral ∫x cos x/ sin dx = (1/2)x² - (1/sin x) + C + C', where C and C' are constants of integration.