Mastering Integrals: How to Find the Integral of sin²(x) Using Trigonometric Identities and Integration Techniques

∫sin²(x)dx

To find the integral of sin²(x), we can use a trigonometric identity and integration techniques

To find the integral of sin²(x), we can use a trigonometric identity and integration techniques.

Step 1: Using the double-angle identity for sine, we can rewrite sin²(x) as (1/2)(1 – cos(2x)).

Step 2: Now we have ∫(1/2)(1 – cos(2x)) dx.

Step 3: Distribute the (1/2) to get (∫(1/2)dx) – (∫(1/2)cos(2x)dx).

Step 4: The integral of (1/2) with respect to x is (1/2)x.

Step 5: To integrate cos(2x), we can use a substitution. Let u = 2x, then du = 2dx.

Step 6: Rearranging, we have dx = (1/2)du.

Step 7: Substitute dx and cos(2x) in the integral, giving us -(1/2)∫cos(u)du.

Step 8: The integral of cos(u) with respect to u is sin(u), so -(1/2)∫cos(u)du becomes -(1/2)sin(u).

Step 9: Substitute back u = 2x, giving us -(1/2)sin(2x).

Step 10: Putting it all together, we have ∫sin²(x)dx = (1/2)x – (1/2)sin(2x) + C, where C is the constant of integration.

Thus, the integral of sin²(x) is (1/2)x – (1/2)sin(2x) + C.

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