What is the integral of #(secxtanx)(1+secx)dx#?
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To find the integral of ((\sec(x)\tan(x))(1+\sec(x)),dx), first expand the expression inside the integral:
[ \int (\sec(x)\tan(x))(1+\sec(x)),dx = \int (\sec(x)\tan(x) + \sec^2(x)\tan(x)),dx ]
Now, integrate each term separately:

For (\int \sec(x)\tan(x),dx), we know the integral of (\sec(x)\tan(x)) is (\sec(x)). This is a standard result from integration formulas.

For (\int \sec^2(x)\tan(x),dx), let's use a substitution method. Let (u = \sec(x)), which implies (du = \sec(x)\tan(x),dx). The integral becomes:
[ \int u,du = \frac{1}{2}u^2 = \frac{1}{2}\sec^2(x) ]
So, combining both parts, the integral is:
[ \sec(x) + \frac{1}{2}\sec^2(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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