Integration by Trigonometric Substitution Calculator

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Example

Created on 2024-06-20Asked by Ella Anderson (Solvelet student)

Evaluate the integral

\int \frac{dx}{\sqrt{4 - x^2}}

using trigonometric substitution.

Solution

To evaluate the integral

\int \frac{dx}{\sqrt{4 - x^2}}

using trigonometric substitution: Let

x = 2 \sin \theta

dx = 2 \cos \theta \, d\theta

Substitute

x

and

dx

\int \frac{dx}{\sqrt{4 - x^2}} = \int \frac{2 \cos \theta \, d\theta}{\sqrt{4 - 4 \sin^2 \theta}}

= \int \frac{2 \cos \theta \, d\theta}{\sqrt{4(1 - \sin^2 \theta)}}

= \int \frac{2 \cos \theta \, d\theta}{2 \cos \theta}

= \int d\theta

Integrate

\theta

\theta + C

Now, substitute back

\theta = \sin^{-1} \left( \frac{x}{2} \right)

\sin^{-1} \left( \frac{x}{2} \right) + C

Therefore, the integral

\int \frac{dx}{\sqrt{4 - x^2}}

is:

\int \frac{dx}{\sqrt{4 - x^2}} = \sin^{-1} \left( \frac{x}{2} \right) + C

Where

C

is the constant of integration. Solved on Solvelet with Basic AI Model

Some of the related questions asked by Oliver Ramirez on Solvelet

1. Evaluate the integral

\int \sqrt{a^2 - x^2} \, dx

using trigonometric substitution,2. Compute the integral

\int x^2 e^x \, dx

using integration by parts.,

DefinitionOne method of evaluating integrals of square roots of quadratic expressions is through the use of trigonometric substitution. Through substitution of a trigonometric function for the variable the integral is transformed into an easier to calculate trigonometric integral. If math has served me correctly, this is again true, but we can work this to our benefit by using clever trigonometry identities to simplify the integrand. Example: To find ∫1−x2dx, use the substitution x=sinθ. Therefore, dx=cosθdθ⇒ 1−x2=1−sin2θ=cosθ. The integral is,∫cos2θdθ (1)This is solvable with trigonometric identities.

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