Weierstrass substitution Calculator

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Example

Created on 2024-06-20Asked by Chloe Moore (Solvelet student)

Use the Weierstrass substitution to evaluate the integral

\int \frac{dx}{\sqrt{x^2 + a^2}}

Solution

To evaluate the integral

\int \frac{dx}{\sqrt{x^2 + a^2}}

using the Weierstrass substitution, we make the substitution

x = a\tan\theta

. 1. **Determine the differential

dx

:** Differentiating both sides of

x = a\tan\theta

with respect to

\theta

, we get:

dx = a\sec^2\theta d\theta.

2. **Substitute for

x

and

dx

:** Substituting

x = a\tan\theta

and

dx = a\sec^2\theta d\theta

into the integral, we get:

\int \frac{a\sec^2\theta d\theta}{\sqrt{a^2\tan^2\theta + a^2}}.

3. **Simplify the integrand:**

\int \frac{a\sec^2\theta d\theta}{\sqrt{a^2\tan^2\theta + a^2}} = \int \frac{a\sec^2\theta d\theta}{\sqrt{a^2(\tan^2\theta + 1)}}.

= \int \frac{a\sec^2\theta d\theta}{\sqrt{a^2\sec^2\theta}} = \int d\theta.

4. **Evaluate the integral:**

\int d\theta = \theta + C.

5. **Convert back to

x

:** Since

x = a\tan\theta

, we have

\tan\theta = \frac{x}{a}

. Therefore, the final result is:

\int \frac{dx}{\sqrt{x^2 + a^2}} = \arctan\left(\frac{x}{a}\right) + C.

Solved on Solvelet with Basic AI Model

Some of the related questions asked by William Hill on Solvelet

1. Use the Weierstrass substitution to evaluate the integral

\int \frac{1+\sin(x)}{1} dx

.2. Simplify the integral

\int \frac{1 + \tan^2(x)}{dx}

using the Weierstrass substitution.,

DefinitionThe Weierstrass substitution is a method used for the reduction of integrals involving trigonometry to integrals involving rational functions. It converts the trigonometric expressions into rational ones by using the trigonometric identities. Illustration: Replace t=tan(2x) ⇒ sin(x)dx becomes equal to 1+t22dt.

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