# Solving Integral of cos^2*sqrt(u)

• Punkyc7
In summary, the integral of cos^2(x)*sqrt(1+tan^2(x)) can be simplified using trigonometric identities and then easily integrated. The substitution u = 1 + tan^2(x) is not a good choice, and instead, a different approach should be used.
Punkyc7

## Homework Statement

integral of cos^2 * (sqr(1+tan^2))

## The Attempt at a Solution

let u = 1 +tan^2
du = sec^2

im not sure how 1/cos^2 gets accounted for the cos^2 in the front

You did your differentiation wrong. In fact you don't need to use substitution. Firstly what is 1+tan^{2}x?

Punkyc7 said:

## Homework Statement

integral of cos^2 * (sqr(1+tan^2))

## The Attempt at a Solution

let u = 1 +tan^2
du = sec^2
If u = 1 + tan^2(theta), du = 2tan(theta)*sec^2(theta)d(theta)

In any case, this is not a good substitution to use.

There are some trig identities that can be used to simplify the integrand first, and then the integration is very simple.
Punkyc7 said:
im not sure how 1/cos^2 gets accounted for the cos^2 in the front

## 1. What is the process for solving the integral of cos^2*sqrt(u)?

The process for solving this integral involves using a trigonometric identity to rewrite cos^2*sqrt(u) as a function of u, and then applying the power rule for integrals to solve for the antiderivative.

## 2. How do I use a trigonometric identity to rewrite cos^2*sqrt(u)?

You can use the identity cos^2(x) = (1 + cos(2x))/2 to rewrite cos^2*sqrt(u) as (1 + cos(2*sqrt(u)))/2. Then, substitute u for x to get the final form of the integral.

## 3. Can this integral be solved using substitution?

Yes, this integral can be solved using substitution. You can substitute u = x^2 and then use the chain rule to rewrite the integral in terms of u. However, using a trigonometric identity may be a simpler and more direct approach.

## 4. Is there a specific range of values for u that this integral can be solved for?

No, there is no specific range of values for u that this integral can be solved for. It can be solved for any value of u, as long as you use the appropriate trigonometric identity and apply the power rule for integrals correctly.

## 5. Are there any special techniques or tips for solving this integral?

One helpful tip for solving this integral is to use the half-angle formula for cosine, which states that cos^2(x) = (1 + cos(2x))/2. This can make the integration process simpler and more straightforward.

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