# Integrate sinx*sqrt(1+((cosx)^2))dx

• Maiko
In summary, the conversation discusses various approaches to solving the integral of sinx*sqrt(1+((cosx)^2))dx, including using integration by parts and various substitutions such as tan(x) = t and cos(x) = t. One suggestion is to use the substitution u = cosx, which simplifies the integral to -\int \sqrt[]{1+u^2} du.
Maiko

## Homework Statement

Integrate sinx*sqrt(1+((cosx)^2))dx

## Homework Equations

integral udv = uv - integral vdu

## The Attempt at a Solution

I tried integration by parts which is: integral udv = uv - integral vdu
I tried substituting (cosx)^2= 1-(sonx)^2
neither of them seemed to work...

Hi, the your integrated is resolved by wolframalpha.com.
Here's the image:

<< Complete solution removed by Hootenanny >>

Use the substitution tan(x) = t

$$sin^2(x)=\frac{tan^2(x)}{1+tan^2(x)}=\frac{t^2}{1+t^2}$$

$$cos^2(x)=\frac{1}{1+tan^2(x)}=\frac{1}{1+t^2}$$

$$dx=\frac{dt}{1+t^2}$$

Or even better, you can use the substitution cos(x)=t

You could also use the substitution u = cosx. Then your integral turns into $$- \int \sqrt[]{1+u^2} du$$.

Do you recognize this integral now?

## 1. What is the formula for integrating sinx*sqrt(1+((cosx)^2))dx?

The formula for integrating sinx*sqrt(1+((cosx)^2))dx is:
∫sinx*sqrt(1+((cosx)^2))dx = -√(1+((cosx)^2)) + C

## 2. How do you solve the integral of sinx*sqrt(1+((cosx)^2))dx?

To solve the integral of sinx*sqrt(1+((cosx)^2))dx, you can use the substitution method by letting u = cosx. This will simplify the integral to:
∫sinx*sqrt(1+((cosx)^2))dx = ∫sqrt(1+u^2)du = (1/2)*(u*sqrt(1+u^2) + ln|u+sqrt(1+u^2)|) + C
Substituting back for u = cosx, the final solution is:
∫sinx*sqrt(1+((cosx)^2))dx = -√(1+((cosx)^2)) + (1/2)*(cosx*sqrt(1+(cosx)^2) + ln|cosx+sqrt(1+(cosx)^2)|) + C

## 3. What are the steps for solving the integral of sinx*sqrt(1+((cosx)^2))dx?

The steps for solving the integral of sinx*sqrt(1+((cosx)^2))dx are:
1. Use the substitution method by letting u = cosx
2. Simplify the integral to ∫sqrt(1+u^2)du
3. Integrate using the formula for ∫sqrt(1+x^2)dx = (1/2)*(x*sqrt(1+x^2) + ln|x+sqrt(1+x^2)|) + C
4. Substitute back for u = cosx
5. Simplify and solve for the final solution.

## 4. Can you use any other methods to solve the integral of sinx*sqrt(1+((cosx)^2))dx?

Yes, you can also use the trigonometric identity sin^2x + cos^2x = 1 to rewrite the integral as:
∫sinx*sqrt(1+((cosx)^2))dx = ∫sinx*sqrt(sin^2x + 1)dx
Then, use the substitution method by letting u = sinx and simplify the integral to:
∫sqrt(u^2 + 1)du = (1/2)*(u*sqrt(u^2 + 1) + ln|u+sqrt(u^2 + 1)|) + C
Substituting back for u = sinx, the final solution is:
∫sinx*sqrt(1+((cosx)^2))dx = (1/2)*(sinx*sqrt(sin^2x + 1) + ln|sinx+sqrt(sin^2x + 1)|) + C

## 5. What is the importance of integrating sinx*sqrt(1+((cosx)^2))dx in scientific applications?

The integral of sinx*sqrt(1+((cosx)^2))dx has various applications in physics, engineering, and other scientific fields. It can be used to calculate the arc length of a curve, the surface area of a solid of revolution, and the work done by a force along a curved path. It is also used in signal processing and image processing to filter and analyze data. Overall, the integral has a wide range of applications and is an important concept in the study of mathematics and science.

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