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Trigonometric Integrals Calculator

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Example
Created on 2024-06-20Asked by Elizabeth Carter (Solvelet student)
Evaluate the integral ∫sin⁡(x)cos⁡(x) dx\int \sin(x) \cos(x) \, dx.

Solution

To verify the identity sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1: 1. **Pythagorean identity:** sin⁡2(x)+cos⁡2(x)=1. \sin^2(x) + \cos^2(x) = 1. 2. **Using the unit circle:** - On the unit circle, any point (cos⁡(x),sin⁡(x))(\cos(x), \sin(x)) lies on the circle defined by x2+y2=1x^2 + y^2 = 1. - Therefore, cos⁡2(x)+sin⁡2(x)=1\cos^2(x) + \sin^2(x) = 1. 3. **Result:** The identity sin⁡2(x)+cos⁡2(x)=1\sin^2(x) + \cos^2(x) = 1 is verified. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Mia Young on Solvelet
1. Evaluate the integral ∫sin⁡2(x)dx\int \sin^2(x) dx.2. Find the value of ∫0π2cos⁡(x) dx \int_{0}^{\frac{\pi}{2}} \cos(x) \, dx .,
DefinitionTrigonometric integrals: Integrals of products of trigonometric functions. The use of identities to obtain integrals in a simpler form than the original integrals is one of the techniques. Sample — ∫sin2(x)dx => to integrate, we will integrate with sin2(x)=21−cos(2x) to integrate from the past.
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