The integrals in Exercises 1–44 are in no particular order. Ev...

Question

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.

∫ (1 / (cos² x tan x)) dx from π/3 to π/4

Accepted Answer

Welcome back everyone. Compute the integral from pi divided by 4 up to pi divided by 2 of D X divided by sin squared x tangent x. For this problem, first of all, let's rewrite the integrant. We have our function, which is a 1 divided by sin squared x multiplied by tangent x. So a good idea is to express the ancient in terms of sine and cosine. We're going to get 1 divided by sine squared x, which is multiplied by. Sine x divided by cosine x, right? Because tangent is sine divided by cosine, and now we can divide, right? We can flip the denominator to get cosine x divided by sine cubed x. Notice now that we are integrating. Cosine x divided by. Sign cub XD X. From pi divided by 4 up to pi divided by 2. And we can integrate this function using U substitution if we let you be equal to sin x. And then we know that we want to find the derivative of you with respect to x, which is the derivative of sine, and that's cosine x. If we multiply both sides by dx, we can show that DU is equal to cosine xDX, and this is particularly useful because we have cosine x multiplied by dx within the integral, right? Our next step is to find the new limits of integration, so the lower one is going to be signed off by divided by 4. Which is equal to the square root of 2 divided by 2, and the upper limit of integration is sin of pi divided by 2, which is 1. So now the integral becomes integral from square root of 2 divided by 2 up to 1. Cosine X D X becomes DU. And in the denominator we have U cubed. How do we integrate this function? Well, we want to apply the power rule, but to apply the power rule, we're going to rewrite DU divided by u cubed as u to the power of -3DU. Let's add the limits of integration and now let's integrate. So the integral of U to the power of -3, according to the power rule is U to the power of -2 divided by -2 And we want to evaluate the result from square root of 2 divided by 2 up to 1. We can also simplify this expression by showing that this is negative1 divided by. 2 U squared. Evaluated from square root of 2 divided by 2 up to 1. And now, let's apply the fundamental theorem of calculus, part 2. So we are going to get -1 divided by. 2 multiplied by 12 minus -1 divided by 2 multiplied by the square root of 2 divided by 22. This is going to be equal to negative. One half for the first term and for the second term we're going to add. 1 divided by 2 multiplied by the square root of 2 squared is going to be. 2, and we're dividing by 2 squad which is 4. So, 2 divided by 4 is 1/2, and 2 multiplied by 1/2 is equal to 1, right? One divided by one and gives us one. So we basically get 1/2 plus 1, which is 1/2, and therefore our final answer is 1/2. Thank you for watching.

The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.∫ (1 / (cos² x tan x)) dx from π/3 to π/4

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The integrals in Exercises 1–44 are in no particular order. Evaluate each integral using any algebraic method, trigonometric identity, or substitution you think is appropriate.
∫ (1 / (cos² x tan x)) dx from π/3 to π/4