Find the integral of sinx/cos^3x dx

[tjbateh]

Homework Statement

Find the integral of sinx/cos^3x dx

Homework Equations

The Attempt at a Solution

How would I approach such a problem?

 

[snipez90]

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).

 

[tjbateh]

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

 

[Gregg]

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

 

[tjbateh]

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

 

[slider142]

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

The integrand of an integral is the expression between the summa \int and the differential dx. The equation you wrote above is correct for the integrand, which you can now integrate easily.

 

[Gregg]

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

Yes. If

y=\sec^2(x)


Then:


\frac{dy}{dx} = \cdots

 

[tjbateh]

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.

 

[slider142]

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.

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

 

[Gregg]

y=sec^2(x) \Rightarrow y=\frac{1}{cos(x)}\frac{1}{cos(x)}

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.

 

[tjbateh]

wow, it finally makes sense! Thank you everyone!

 

[njama]

Here is much simpler approach:

\int{\frac{sinx}{cos^3x} dx}

u=cos(x)

du=-sin(x)dx

dx=-du/sin(x)

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

=-\int \frac{du}{u^3}

 

[tjbateh]

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

 

[Bohrok]

You use \ln 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.