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How do you use Integration by Substitution to find #inttan(x)*sec^3(x)dx#?

Answer 1

Emily Ammerman

#=(sec^3(x))/3+c#, where #c# is a constant

Explanation

#=inttan(x)*sec^3(x)dx#

let's have #secx=t#

#sec(x)*tan(x)dx=dt#

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

#=intt^2dt#, which is quite straight forward,

#=t^3/3+c#, where #c# is a constant

#=(sec^3(x))/3+c#, where #c# is a constant

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Answer 2

Elijah Abram

To integrate ( \tan(x) \cdot \sec^3(x) , dx ) using integration by substitution:

Let ( u = \sec(x) ).
Find ( du = \sec(x) \tan(x) , dx ).
Rewrite the integral in terms of ( u ): ( \int u^3 , du ).
Integrate ( u^3 ) to get ( \frac{u^4}{4} ).
Replace ( u ) with ( \sec(x) ).
The final result is ( \frac{\sec^4(x)}{4} + C ), where ( C ) is the constant of integration.

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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.