Trigonometric integrals

Material by the IntegralPaso project. Conditions and limits are stated in each guide; external teaching review is pending.

In products of sine and cosine, an odd power lets you separate a factor and use substitution. For even powers, double-angle identities can reduce the degree. Roots such as √(a²−x²) suggest a trigonometric substitution, but an interval must be specified to control signs and roots.

Worked example

1. Identify the integrand, variables and order. For multiple integrals, start with the innermost integral.

   $$\int \sin^{2}{\left(x \right)}\, dx$$

2. Integrate with respect to x. Treat the other variables as constants at this stage.

   $$\int \sin^{2}{\left(x \right)}\, dx$$

3. Look for trigonometric identities that simplify the integrand, then apply integration rules to the resulting expression.

   $$\sin^{2}{\left(x \right)} = \sin^{2}{\left(x \right)}$$

4. Rewrite the expression in an equivalent form that is easier to integrate.

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

5. Split the sum. The integral of a sum is the sum of its integrals.

   $$\int \left(\frac{1}{2} - \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = \int \frac{1}{2}\, dx + \int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx$$

6. Integrate a constant by multiplying it by the variable.

   $$\int \frac{1}{2}\, dx = \frac{x}{2}$$

7. Move the constant factor outside: it does not depend on the integration variable.

   $$\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\sin{\left(2 x \right)}}{4}$$

8. Make a substitution and replace the differential too; both must change together.

   $$u = 2 x,\quad du = 2\,dx$$

9. Move the constant factor outside: it does not depend on the integration variable.

   $$\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\sin{\left(u \right)}}{2}$$

10. Use the fact that the derivative of sin(x) is cos(x).

    $$\int \cos{\left(u \right)}\, du = \sin{\left(u \right)}$$

11. Combine the results of the substeps and simplify.

    $$\int \frac{\cos{\left(u \right)}}{2}\, du = \frac{\sin{\left(u \right)}}{2}$$

12. After integrating the expression in u, return to the original variable.

    $$\int \cos{\left(2 x \right)}\, dx = \frac{\sin{\left(2 x \right)}}{2}$$

13. Combine the results of the substeps and simplify.

    $$\int \left(- \frac{\cos{\left(2 x \right)}}{2}\right)\, dx = - \frac{\sin{\left(2 x \right)}}{4}$$

14. Combine the results of the substeps and simplify.

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

15. Combine the results of the substeps and simplify.

    $$\int \sin^{2}{\left(x \right)}\, dx = \frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4}$$

16. Check this primitive by differentiating it: recover exactly the integrand of this stage.

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

17. Add the constant C: all primitives on an interval differ by a constant.

    $$\frac{x}{2} - \frac{\sin{\left(2 x \right)}}{4} + C$$