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Integrals with Radicals Calculator

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Example
Created on 2024-06-20Asked by Emma Allen (Solvelet student)
Evaluate the integral ∫4−x2 dx \int \sqrt{4 - x^2} \, dx .

Solution

To evaluate the integral ∫4−x2 dx \int \sqrt{4 - x^2} \, dx : Use the trigonometric substitution x=2sin⁡θ x = 2 \sin \theta : dx=2cos⁡θ dθ dx = 2 \cos \theta \, d\theta Substitute x x and dx dx : ∫4−x2 dx=∫4−(2sin⁡θ)2⋅2cos⁡θ dθ \int \sqrt{4 - x^2} \, dx = \int \sqrt{4 - (2 \sin \theta)^2} \cdot 2 \cos \theta \, d\theta =∫4−4sin⁡2θ⋅2cos⁡θ dθ = \int \sqrt{4 - 4 \sin^2 \theta} \cdot 2 \cos \theta \, d\theta =∫4(1−sin⁡2θ)⋅2cos⁡θ dθ = \int \sqrt{4(1 - \sin^2 \theta)} \cdot 2 \cos \theta \, d\theta =∫21−sin⁡2θ⋅2cos⁡θ dθ = \int 2 \sqrt{1 - \sin^2 \theta} \cdot 2 \cos \theta \, d\theta =2∫cos⁡2θ dθ = 2 \int \cos^2 \theta \, d\theta Use the identity cos⁡2θ=12(1+cos⁡2θ) \cos^2 \theta = \frac{1}{2}(1 + \cos 2\theta) : =2∫12(1+cos⁡2θ) dθ = 2 \int \frac{1}{2}(1 + \cos 2\theta) \, d\theta =∫(1+cos⁡2θ) dθ = \int (1 + \cos 2\theta) \, d\theta =∫dθ+∫cos⁡2θ dθ = \int d\theta + \int \cos 2\theta \, d\theta =θ+12sin⁡2θ+C = \theta + \frac{1}{2} \sin 2\theta + C Now, substitute x=2sin⁡θ x = 2 \sin \theta : =sin⁡−1(x2)+12sin⁡(2sin⁡−1(x2))+C = \sin^{-1} \left( \frac{x}{2} \right) + \frac{1}{2} \sin \left( 2 \sin^{-1} \left( \frac{x}{2} \right) \right) + C Therefore, the integral ∫4−x2 dx \int \sqrt{4 - x^2} \, dx is: ∫4−x2 dx=sin⁡−1(x2)+12sin⁡(2sin⁡−1(x2))+C \int \sqrt{4 - x^2} \, dx = \sin^{-1} \left( \frac{x}{2} \right) + \frac{1}{2} \sin \left( 2 \sin^{-1} \left( \frac{x}{2} \right) \right) + C Where C C is the constant of integration. Solved on Solvelet with Basic AI Model
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1. Evaluate the integral ∫02x2+1 dx\int_0^2 \sqrt{x^2 + 1} \, dx,2. Solve the first-order linear differential equation dydx+1xy=x2\frac{dy}{dx} + \frac{1}{x}y = x^2 using the integrating factor method.,
DefinitionThe integrals with radicals are functions that have a square root or a root for which we search the antiderivative. These kinds of integrals typically arise from a combination of substitution, trigonometric identities, or rationalization tricks to simplify the integrand. This has applications in geometry, physics, mechanics among many others. For Example ∫x​dx will use Power Rule for Integration and simplify to ∫x1/2dx=32​x3/2+C
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