Recomposition Math Example 4

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Example 4

medium
Given the identity sin⁡2θ=1−cos⁡2θ2\sin^2\theta = \frac{1-\cos 2\theta}{2}, recompose: use it to evaluate ∫sin⁡2θ dθ\int \sin^2\theta\, d\theta.

Solution

  1. 1
    Replace sin⁡2θ\sin^2\theta with 1−cos⁡2θ2\frac{1-\cos 2\theta}{2}: ∫sin⁡2θ dθ=∫1−cos⁡2θ2 dθ\int \sin^2\theta\,d\theta = \int \frac{1-\cos 2\theta}{2}\,d\theta.
  2. 2
    Integrate: =12θ−sin⁡2θ4+C= \frac{1}{2}\theta - \frac{\sin 2\theta}{4} + C.
  3. 3
    Recompose the result back in terms of θ\theta: θ2−sin⁡2θ4+C\frac{\theta}{2} - \frac{\sin 2\theta}{4} + C.

Answer

∫sin⁡2θ dθ=θ2−sin⁡2θ4+C\int \sin^2\theta\,d\theta = \frac{\theta}{2} - \frac{\sin 2\theta}{4} + C
Recomposition here means applying a known identity to rewrite sin⁡2θ\sin^2\theta in an integrable form. The identity was decomposed to a simpler expression; after integrating, the result is the recomposed answer.

About Recomposition

Recomposition is the process of combining simpler parts, sub-results, or solved sub-problems back together to form a complete solution or to understand the whole structure from its pieces.

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More Recomposition Examples

Example 1 easy

After decomposing [formula], recompose to solve [formula].

Example 2 medium

Partial fractions gave [formula]. Recompose to verify by adding the fractions.

Example 3 easy

You find [formula] and [formula] from solving a system. Recompose by substituting back into both equ

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Related Concepts

Decomposition