What is #int tan^3(2x) sec^100(2x) dx#?

Answer 1

Emilia Aguilar

#1/204*sec^102 (2x) -1/200*sec^100 (2x) + C#

Reduce the integral #int tan ^3(2x) sec^100 (2x) ##dx# into

#int sec^99 2x* tan^2 2x*(sec 2x* tan 2x) # #dx#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Aurora Alvarado

To integrate ( \tan^3(2x) \sec^{100}(2x) , dx ), we can use the substitution method:

Let ( u = \tan(2x) ), then ( du = 2\sec^2(2x) , dx ).

This means that ( dx = \frac{1}{2\sec^2(2x)} , du = \frac{1}{2(u^2 + 1)} , du ).

Substituting these into the integral:

[ \int \tan^3(2x) \sec^{100}(2x) , dx = \int u^3 \sec^{100}(2x) \cdot \frac{1}{2(u^2 + 1)} , du ]

[ = \frac{1}{2} \int \frac{u^3}{(u^2 + 1)^{100}} , du ]

Now, you can integrate this using standard techniques, such as partial fractions or possibly another method depending on the complexity of the integral.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.