# Integration by substitution ((sin(x))/(1+cos^2(x)))dx

• sapiental
In summary, the conversation discusses how to evaluate the indefinite integral ((sin(x))/(1+cos^2(x)))dx. Two different methods are suggested, with one involving setting u = 1 + cos^2(x) and the other involving setting u = cos(x). There is also a discussion on how to format equations in the message board for better clarity.
sapiental
evaluat the indefinite integral ((sin(x))/(1+cos^2(x)))dx

I let

u = 1 + cos^2(x)

then du = -sin^2(x)dx

I rewrite the integral to

- integral sqrt(du)/u

can I set it up like this? should I change u to something else?

I also tried it like this by rewriting the original equation to:

indefinite integral ((sin(x))/(1+cos(x)cos(x)))dx

u = cos(x)

du = -sin(x)dx

then

- integral (du)/(1+(u^2))

Also, can somebody give me directions on how to format equations in this message board to make my questions somewhat clearer.

Thanks alot!

sapiental said:
evaluat the indefinite integral ((sin(x))/(1+cos^2(x)))dx

I let

u = 1 + cos^2(x)

then du = -sin^2(x)dx

I rewrite the integral to

- integral sqrt(du)/u

That's not right. The derivative of $$\cos^2{x}$$ is NOT $$-\sin^2{x}$$.

can I set it up like this?

I also tried it like this by rewriting the original equation to:

indefinite integral ((sin(x))/(1+cos(x)cos(x)))dx

u = cos(x)

du = -sin(x)dx

then

- integral (du)/(1+(u^2))

Yes you can. The final integral is pretty straightforward.

Also, can somebody give me directions on how to format equations in this message board to make my questions somewhat clearer.

neutrino said:
That's not right. The derivative of $$\cos^2{x}$$ is NOT $$-\sin^2{x}$$.
Yes you can. The final integral is pretty straightforward.

neutrino's right - you can't differentiate $$\cos^2{x}$$ as $$-\sin^2{x}$$. If it helps, think of $$\cos^2{x}$$ as $$cos{x} * cos{x}$$. You can then use the chain rule.

## 1. What is integration by substitution?

Integration by substitution is a method used in calculus to evaluate integrals. It involves replacing the variable of integration with a new variable in order to simplify the integral and make it easier to solve.

## 2. How is integration by substitution used to solve ((sin(x))/(1+cos^2(x)))dx?

To solve this integral, we can use the substitution u = cos(x). This will allow us to rewrite the integral as ((sin(x))/(1+cos^2(x)))dx = ((sin(x))/(1+u^2))du. We can then use trigonometric identities to simplify the integral and solve for u. Once we have the solution in terms of u, we can substitute back in cos(x) for u to get the final answer.

## 3. What are the steps to perform integration by substitution?

The steps to perform integration by substitution are as follows:1. Identify a suitable substitution for the variable of integration.2. Rewrite the integral in terms of the new variable.3. Use algebraic or trigonometric identities to simplify the integral.4. Solve the integral in terms of the new variable.5. Substitute back in the original variable to get the final answer.

## 4. What is the purpose of using integration by substitution?

The purpose of using integration by substitution is to simplify the integral and make it easier to solve. This method allows us to change the variable of integration to one that is more manageable, leading to a simpler and more straightforward solution.

## 5. Are there any tips for choosing a suitable substitution for integration by substitution?

Yes, there are a few tips for choosing a suitable substitution:- Look for expressions that appear multiple times in the integral.- Choose a substitution that will eliminate a complicated or difficult expression.- Use trigonometric identities to rewrite the integral in terms of a simpler trigonometric function.- If the integral contains a radical, consider substituting for the entire expression under the radical.

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