# What is the Integral of dx/ (1+cos ^2(x)) Using Different Approaches?

• belleamie
In summary, the formula for calculating the integral of dx/ (1+cos ^2(x)) is ∫ dx/ (1+cos ^2(x)) = tan(x) + C, where C is the constant of integration. To solve this integral, the substitution method can be used by substituting u = cos(x) and du = -sin(x) dx. The domain and range of this integral are both all real numbers. It can also be used to find the area under a curve by taking the definite integral between two given points. Additionally, the integral can be solved using the trigonometric identity cos^2(x) = (1+cos(2x))/2.
belleamie
for the life of me i can't seem to understand how to the the intergral of dx/ (1+cos ^2(x))?

Use a substitution

$$\tan\frac{x}{2}=t$$

and some trigonometry.

Daniel.

Daniel's approach probably works equally well; here's another approach:
$$\frac{1}{1+\cos^{2}x}=\frac{1}{\cos^{2}x}\frac{1}{1+\frac{1}{\cos^{2}x}}=(\frac{d}{dx}tan(x))\frac{1}{2+\tan^{2}x}$$
Thus, setting $$u=tan(x)$$, we have $$\frac{du}{dx}dx=du$$, that is:
$$\int\frac{dx}{1+\cos^{2}x}=\int\frac{du}{2+u^{2}}$$

## 1. What is the formula for calculating the integral of dx/ (1+cos ^2(x))?

The formula for calculating the integral of dx/ (1+cos ^2(x)) is ∫ dx/ (1+cos ^2(x)) = tan(x) + C, where C is the constant of integration.

## 2. How do you solve the integral of dx/ (1+cos ^2(x))?

To solve the integral of dx/ (1+cos ^2(x)), you can use the substitution method by substituting u = cos(x) and du = -sin(x) dx. This will transform the integral into ∫ du/ (1+u^2), which can be solved using the arctan formula.

## 3. What is the domain and range of the integral of dx/ (1+cos ^2(x))?

The domain of the integral of dx/ (1+cos ^2(x)) is all real numbers, while the range is also all real numbers.

## 4. Can the integral of dx/ (1+cos ^2(x)) be used to find the area under a curve?

Yes, the integral of dx/ (1+cos ^2(x)) can be used to find the area under a curve. It represents the antiderivative of the function, and the definite integral can be used to find the area under the curve between two given points.

## 5. Are there any other methods to solve the integral of dx/ (1+cos ^2(x))?

Yes, apart from the substitution method, the integral of dx/ (1+cos ^2(x)) can also be solved using the trigonometric identity cos^2(x) = (1+cos(2x))/2. This will transform the integral into a simpler form that can be solved using basic integration techniques.

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