- #1

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## Homework Statement

Find the integral of sinx/cos^3x dx

## Homework Equations

## The Attempt at a Solution

How would I approach such a problem?

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- #1

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Find the integral of sinx/cos^3x dx

How would I approach such a problem?

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- #2

- 1,101

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If that's a cos(x) raised to the 3rd power in the denominator, then your integrand is just tan(x)sec^2(x) which is easy since the derivative of tan(x) is sec^2(x).

- #3

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Wait so if I substitute U for cos(x)^3, then du=tan(x)sec^2(x)?

- #4

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No, he's saying that the integrand can be rewritten as tanxsec^2 x which can be easily integrated.

- #5

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the integrand as in the whole problem? So sin(x)/cos(x)^3=tan(x)sec(x)^2??

- #6

- 1,013

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The integrand of an integral is the expression between the summa [tex]\int[/tex] and the differential dx. The equation you wrote above is correct for the integrand, which you can now integrate easily.the integrand as in the whole problem? So sin(x)/cos(x)^3=tan(x)sec(x)^2??

- #7

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Yes. Ifthe integrand as in the whole problem? So sin(x)/cos(x)^3=tan(x)sec(x)^2??

[tex]y=\sec^2(x)[/tex]

Then:

[tex]\frac{dy}{dx} = \cdots [/tex]

- #8

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tan(x)sec(x)?? But i don't understand, what happens to the sin(x) in the numerator..It seems like were just talking about the denominator.

- #9

- 1,013

- 70

Do you remember the common definition of tan(x) = sin(x)/cos(x)?

- #10

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[tex] y=sec^2(x) \Rightarrow y=\frac{1}{cos(x)}\frac{1}{cos(x)}[/tex]

Use the product rule to differentiate that, you will see you have your derivative that is the [almost] the same as the integrand. Hence you have the answer.

- #11

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wow, it finally makes sense! Thank you everyone!

- #12

- 216

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Here is much simpler approach:

[tex]\int{\frac{sinx}{cos^3x} dx}[/tex]

u=cos(x)

du=-sin(x)dx

dx=-du/sin(x)

[tex]\int{\frac{sin(x)}{u^3}*\frac{-du}{sin(x)}}=[/tex]

[tex]=-\int \frac{du}{u^3}[/tex]

- #13

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This was more of the approach we learned in class. Would you then use the LN function?

- #14

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You use [itex]\ln[/itex] if the integrand were 1/u where the denominator has a power of one, but for any other power, use the power rule for integrals.

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