Evaluating the integral with trig

• m0gh
In summary, when integrating functions, it is important to consider their domains and use absolute value bars if necessary to ensure the correct answer. The natural logarithm, ln, is a logarithm with e as the base and is shorter to write. However, its domain is restricted to positive x values, so when integrating functions with different domains, absolute value bars may be necessary to get the correct answer.
m0gh

Homework Statement

Evaluate the integral x^(-1/5) tan(x^(4/5)) dx

The Attempt at a Solution

Set u = x^(4/5)
Therefore du = 4/5 x ^ (-1/5) dx

which gave me 5/4 ∫ tan(u) du

integral of tan is natural log of absolute value of sec(u)

sub in x^(4/5) for u and get 5/4 natural log of absolute value of sec(x^(4/5)) + c

I'm sorry if I didn't use the proper formatting as I'm not %100 how. I will attach an image to clarify my work. I used an indefinite integral calculator to check my work but it is giving a different answer. Can anyone verify that this solution is correct?

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m0gh said:

Homework Statement

Evaluate the integral x^(-1/5) tan(x^(4/5)) dx

The Attempt at a Solution

Set u = x^(4/5)
Therefore du = 4/5 x ^ (-1/5) dx

which gave me 5/4 ∫ tan(u) du

integral of tan is natural log of absolute value of sec(u)

sub in x^(4/5) for u and get 5/4 natural log of absolute value of sec(x^(4/5)) + c

I'm sorry if I didn't use the proper formatting as I'm not %100 how. I will attach an image to clarify my work. I used an indefinite integral calculator to check my work but it is giving a different answer. Can anyone verify that this solution is correct?

##\ln{|\sec{x}|} = \ln{|\cos{x}^{-1}|} = -\ln{|\cos{x}|}##

Did thje online calculator have the answer in terms of cosine?

Also can someone please explain why in class my teacher uses the natural log of the absolute value when integrating 1/x but when I use something like wolframalpha it always uses log. I have never had a teacher really explain the difference between these (I'm assuming they are interchangeable) and it would really help with my understanding. I've read online but I still don't truly understand what it means.

m0gh said:
Also can someone please explain why in class my teacher uses the natural log of the absolute value when integrating 1/x but when I use something like wolframalpha it always uses log. I have never had a teacher really explain the difference between these (I'm assuming they are interchangeable) and it would really help with my understanding. I've read online but I still don't truly understand what it means.

Here is a better link: http://www.wolframalpha.com/input/?i=integral+of+x^(-1/5)+tan(x^(4/5))

Underneath the answer, it says that log(x) is the natural logarithm. if it was a different base, it wouldn't be written as ##\log_2{x}##, for example. http://www.wolframalpha.com/input/?i=log_2(x)

Right under the example, you can see the general formula for writing the log in any base in terms of the natural logarithm: ##\log_b{x}=\frac{\log{x}}{\log{b}}##. Note what this becomes when b = e.

The natural logarithm has a plethora of use in mathematics, more so than other bases, because of its relationship with e. In practice, it is shorter to write ln than log, so that may be why it is used.

To answer you other question one post up, they are equivalent. Read my other post to see why. When using online calculators, it is important to realize that there may be alternate forms to the answer.

Another hint would be to subtract your answer from the answer WolframAlpha gives to see if the result is 0. http://www.wolframalpha.com/input/?i=5/4*Log|Sec[x^(4/5)]|+5/4*+Log|Cos[x^(4/5)]|

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Okay, so now I understand that ln is just a logarithm with e as the base. What I don't get is the need for the absolute value bars that show up. Is this in the nature of a logarithm? (I don't understand logarithms very well at all.)

m0gh said:
Okay, so now I understand that ln is just a logarithm with e as the base. What I don't get is the need for the absolute value bars that show up. Is this in the nature of a logarithm? (I don't understand logarithms very well at all.)

You know the derivative of ##\ln x## is ##\frac 1 x##. The domain of ##\ln x## is ##x>0##. The domain of ##\frac 1 x## is ##x\ne 0##. So here the logarithm and its derivative seem to have different domains. If we use absolute values, then ##\ln|x|## has domain ##x\ne 0##, and it agrees with ##\ln x## when ##x>0##. What is the derivative of ##\ln|x|## if ##x<0##? Remembering that ##|x| = -x## if ##x<0##, we get$$\frac d {dx}\ln |x| = \frac d {dx}\ln(-x) = \frac 1 {-x}\cdot (-1) = \frac 1 x$$by the chain rule if ##x<0##. So for all ##x\ne 0## the derivative of ##\ln |x|## is ##\frac 1 x##. The absolute value bars are often ignored, especially if the function given is ##\ln x## since its domain is automatically ##x>0##. Taking antiderivatives though, one should use the absolute values unless something in the problem specifies only positive ##x##.

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1. What is the purpose of evaluating integrals with trigonometric functions?

Evaluating integrals with trigonometric functions allows us to solve problems involving circular motion, periodic functions, and other situations where trigonometric functions are present. It also helps us find areas and volumes of irregular shapes and objects.

2. What are the basic trigonometric functions used in evaluating integrals?

The basic trigonometric functions used in evaluating integrals are sine, cosine, and tangent. These functions can be expressed in terms of other trigonometric functions such as secant, cosecant, and cotangent.

3. How do I determine which trigonometric substitution to use when evaluating an integral?

The substitution used in evaluating an integral will depend on the form of the integrand. For integrals involving expressions of the form ax^2 + bx + c, the substitution used will involve the tangent function. For integrals involving expressions of the form x^2 - a^2, the substitution used will involve the secant function. It is important to choose the appropriate substitution to simplify the integral and make it easier to evaluate.

4. What are the common techniques used in evaluating integrals with trigonometric functions?

The common techniques used in evaluating integrals with trigonometric functions include trigonometric substitution, integration by parts, and partial fraction decomposition. These techniques help us simplify complex integrands and make them easier to evaluate using basic integration rules.

5. How can I check if my answer to an integral with trigonometric functions is correct?

To check if your answer to an integral with trigonometric functions is correct, you can differentiate your answer and see if it matches the original integrand. You can also use online tools or calculators to verify your answer. It is important to double-check your work to ensure accuracy.

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