# How do you evaluate the definite integral #int sec^2x/(1+tan^2x)# from #[0, pi/4]#?

Interestingly enough, we can also note that this fits the form of the arctangent integral, namely:

Adding the bounds:

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Notice that from the second Pythagorean identity that

This means the fraction is equal to 1 and this leaves us the rather simple integral of

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To evaluate the definite integral ( \int_{0}^{\frac{\pi}{4}} \frac{\sec^2(x)}{1+\tan^2(x)} , dx ) from (0) to (\frac{\pi}{4}), we can use the trigonometric identity (1 + \tan^2(x) = \sec^2(x)). Therefore, the integral becomes ( \int_{0}^{\frac{\pi}{4}} \frac{\sec^2(x)}{\sec^2(x)} , dx = \int_{0}^{\frac{\pi}{4}} dx ). Integrate (dx) with respect to (x) from (0) to (\frac{\pi}{4}) to obtain the result.

The integral ( \int_{0}^{\frac{\pi}{4}} dx ) is simply the difference of the upper limit and the lower limit, which is ( \frac{\pi}{4} - 0 = \frac{\pi}{4} ). Therefore, the value of the definite integral ( \int_{0}^{\frac{\pi}{4}} \frac{\sec^2(x)}{1+\tan^2(x)} , dx ) from (0) to (\frac{\pi}{4}) is ( \frac{\pi}{4} ).

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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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