How do you find the integral of #(1+ tan^2x)sec^2xdx#?

Answer 1
#int(1+tan^2(x))sec^2(x)dx#

We know :

#1+tan^2(x) = 1/cos^2(x)#
#sec^2(x) = 1/cos^2(x)#
So we have : #int1/cos^4(x)dx#
Let's #t = tan(x) and dt = 1/cos^2(x)dx#

We have :

#int1/cos^2(x)dt#
#int1+tan^2dt#
#int1+t^2dt#
#[t+1/3t^3]#
#[tan(x)+1/3tan^3(x)]+C#
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Answer 2

To find the integral of (1 + tan^2(x))sec^2(x)dx, you can use trigonometric identities and integration techniques. Start by replacing tan^2(x) with sec^2(x) - 1. Then, integrate term by term. Here are the steps:

  1. Rewrite the integral as ∫(1 + sec^2(x) - 1)sec^2(x)dx.
  2. Distribute sec^2(x) to get ∫(sec^2(x) + sec^4(x) - sec^2(x))dx.
  3. Simplify to get ∫(sec^2(x) + sec^4(x))dx.
  4. Integrate each term separately: ∫sec^2(x)dx + ∫sec^4(x)dx.
  5. The integral of sec^2(x) is tan(x) + C, where C is the constant of integration.
  6. To integrate sec^4(x), use a substitution or integration by parts.

Following these steps, the integral of (1 + tan^2(x))sec^2(x)dx can be evaluated.

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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.

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.

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.

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.

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