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Trigonometric Substitution Calculator

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

Solution

To evaluate the integral ∫sin⁡(x)cos⁡(x) dx\int \sin(x) \cos(x) \, dx: 1. **Use the double-angle identity:** sin⁡(x)cos⁡(x)=12sin⁡(2x). \sin(x) \cos(x) = \frac{1}{2} \sin(2x). 2. **Rewrite the integral:** ∫sin⁡(x)cos⁡(x) dx=∫12sin⁡(2x) dx. \int \sin(x) \cos(x) \, dx = \int \frac{1}{2} \sin(2x) \, dx. 3. **Integrate:** ∫12sin⁡(2x) dx=−14cos⁡(2x)+C. \int \frac{1}{2} \sin(2x) \, dx = -\frac{1}{4} \cos(2x) + C. 4. **Result:** The integral ∫sin⁡(x)cos⁡(x) dx\int \sin(x) \cos(x) \, dx is: −14cos⁡(2x)+C. -\frac{1}{4} \cos(2x) + C. Solved on Solvelet with Basic AI Model
Some of the related questions asked by Lucas Adams on Solvelet
1. Use the trigonometric substitution x=atan⁡(θ)x = a \tan(\theta) to evaluate the integral ∫dxx2+a2\int \frac{dx}{x^2 + a^2}.2. Apply trigonometric substitution to find ∫dxa2−x2 \int \frac{dx}{\sqrt{a^2 - x^2}} , where a>0 a > 0 .,
DefinitionTrigonometric substitution (and trig substitution for short) is the name of a technique used in trigonometry (replacing with trigonometric functions) for computing integrals. Example: to measure the weight of ∫a2−x2​dx, you substitute x=asin(θ), and this would give you a simple integral ∫acos2(θ)dθ.
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