How do I evaluate #inttan^3(x) sec^5(x)dx#?
We can write this integral in this way:
I have used the integral:
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To evaluate (\int \tan^3(x) \sec^5(x) , dx):

Use the substitution method:
 Let (u = \tan(x)). Then, (du = \sec^2(x) , dx) and (\sec^2(x) = 1 + \tan^2(x) = 1 + u^2).
 Rewrite the integral in terms of (u).

The integral becomes: [ \int u^3 (1 + u^2)^2 , du ]

Expand and simplify the integrand: [ \int (u^3 + 2u^5 + u^7) , du ]

Integrate each term separately: [ \frac{1}{4}u^4 + \frac{2}{6}u^6 + \frac{1}{8}u^8 + C ]

Substitute back for (u): [ \frac{1}{4}\tan^4(x) + \frac{1}{3}\tan^6(x) + \frac{1}{8}\tan^8(x) + C ]
So, (\int \tan^3(x) \sec^5(x) , dx = \frac{1}{4}\tan^4(x) + \frac{1}{3}\tan^6(x) + \frac{1}{8}\tan^8(x) + C), where (C) is the constant of integration.
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When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
When evaluating a onesided 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.
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