# Two different results depending on integration technique

• awelex
In summary, the conversation is about a person having trouble with an integral and getting two different results depending on the method used. They are confused because Mathematica gives one result while the solution book gives another. However, it is pointed out that both results are correct if the missing constants are included.
awelex
Hi,

I'm having peculiar difficulties with a rather easy integral, namely the integral of -sin(x) / (cos(x))^3. The problem is that depending on which integration technique I choose, I end up with two different result. Moreover, Mathematica gives me one of these two results, while the solution book gives me the other. But the two cannot possibly be the same, unless I'm really overlooking something.

Method One: substitution

u = cos(x)
du = -sin(x) dx

$\int$-sin(x)/cos^3(x) = $\int$u^(-3) = -1/2 u^(-2) = -1/2 sec^2 x

That is also the result that Mathematica gives me

Method Two: Trig Identities, then substitution

-sin x / cos^3 x = -tan x * sec^2 x

u = tan x
du = sec^2 x dx

$\int$-sin(x)/cos^3(x) = - $\int$u = -1/2 u^2 = -1/2 tan^2 x

That's the result that the solution book gives.

It is obvious that the two resulting functions are different. What is wrong here?

Thanks,
Alex

Sorry to say that both results are false because the constants have been forgotten.
If the constants are there, both results are correct and identical :
-(1/2)sec^2 (x) +C1 = -(1/2)tan^2 (x) +C2
C1 = (1/2) + C2

## 1. What is the difference between numerical and analytical integration techniques?

Numerical integration involves approximating the integral using a finite number of numerical evaluations, while analytical integration involves finding the exact solution using mathematical techniques such as substitution, integration by parts, or trigonometric identities.

## 2. Why do different integration techniques give different results?

Integration techniques rely on different mathematical approaches and assumptions, which can lead to variations in the final result. Additionally, numerical integration may introduce errors due to the use of approximations.

## 3. Can one integration technique be considered more accurate than another?

It depends on the specific problem and the desired level of accuracy. In general, analytical integration is considered more accurate as it provides the exact solution, but it may not always be feasible. Numerical integration techniques can also be highly accurate if enough numerical evaluations are used.

## 4. How can I determine which integration technique to use?

The choice of integration technique depends on the complexity of the integral, the available resources (such as time and computing power), and the desired level of accuracy. It is usually a balance between accuracy and efficiency.

## 5. Are there any situations where both analytical and numerical integration techniques are used?

Yes, often a combination of both techniques is used, known as hybrid integration. This involves using analytical techniques for simpler parts of the integral and numerical techniques for more complex parts. This can result in a more accurate and efficient solution.

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