Integrating using trigonometric identities

When studying the integral of $\cos^3(x)$, focus on the following key points:

1. Trigonometric Identity: Use the identity $\cos^2(x) = 1 - \sin^2(x)$ to rewrite $\cos^3(x)$.

2. Expression Rewriting: Rewrite $\cos^3(x)$ as $\cos(x) \cdot (1 - \sin^2(x))$.

3. Substitution: Apply the substitution $u = \sin(x)$, which implies $du = \cos(x) \, dx$.

4. New Integral Form: The integral transforms into a polynomial form in terms of $u$:

   $$\int \cos^3(x) \, dx = \int (1 - u^2) \, du.$$

5. Integration: Compute the integral:

   $$\int (1 - u^2) \, du = u - \frac{u^3}{3} + C.$$

6. Back-substitution: Replace $u$ with $\sin(x)$:

   $$\int \cos^3(x) \, dx = \sin(x) - \frac{\sin^3(x)}{3} + C.$$

7. Result: The final result is:

   $$\int \cos^3(x) \, dx = \frac{3 \sin(x) - \sin^3(x)}{3} + C.$$

These steps provide a structured approach to solving the integral of $\cos^3(x)$.

To study the integral of $\sin^2(x) \cos^3(x)$, focus on the following key points:

1. Trigonometric Identity: Use the identity $\sin^2(x) = 1 - \cos^2(x)$ or similar identities to express the integrand in a more manageable form.

2. Substitution Method: Consider using substitution, such as letting $u = \cos(x)$ or $u = \sin(x)$, to simplify the integral.

3. Differential: Remember to compute the differential $du$ correctly based on the substitution chosen.

4. Power Reduction: If necessary, use power-reducing formulas to simplify $\sin^2(x)$ or $\cos^3(x)$.

5. Integration Techniques: Be familiar with different integration techniques like integration by parts if it becomes applicable.

6. Final Simplification: Don’t forget to convert back to the original variable (if a substitution was applied) and simplify the result.

7. Constants and Limits: If calculating a definite integral, ensure proper evaluation at the limits after integration.

Following these points will help in effectively solving the integral $\int \sin^2(x) \cos^3(x) \, dx$.

When studying the integral of $\sin^4(x)$, there are several key points to focus on:

1. Reduction Formula: Use reduction formulas to express higher powers of sine in terms of simpler integrals.

2. Power-Reduction Formulas: Apply the power-reduction identity:

   $$\sin^2(x) = \frac{1 - \cos(2x)}{2}$$

   to rewrite $\sin^4(x)$ as:

   $$\sin^4(x) = \left(\sin^2(x)\right)^2 = \left(\frac{1 - \cos(2x)}{2}\right)^2$$

3. Expand and Simplify: After substituting, expand the expression:

   $$\sin^4(x) = \frac{1 - 2\cos(2x) + \cos^2(2x)}{4}$$

4. Further Power Reduction: Apply the power-reduction formula again on $\cos^2(2x)$:

   $$\cos^2(2x) = \frac{1 + \cos(4x)}{2}$$

   Substitute this back to simplify the integral further.

5. Integrate Each Term: The final expression will contain simpler integrals involving constant terms and cosines, allowing for straightforward integration.

6. Final Result: Combine and simplify your results to find the final integral of $\sin^4(x)$.

7. Consider the Constant of Integration: Remember to include the constant of integration at the end.

By mastering these steps, you can effectively handle the integral of $\sin^4(x)$.