Use any method to evaluate the integrals in Exercises 65–70.

Question

∫ cot(x) / cos²(x) dx

Accepted Answer

Welcome back, everyone. Evaluate the integral of tangent X divided by cosine squared of X multiplied by D X. For this problem, let's rewrite tangent in terms of its definition. So we're going to have sine of X divided by cosine of X. This is what tangent is, right? And we are dividing it again by cosine squared of X. So now what we're going to do is rewrite our integrant using a single fraction in the numerator we have sine of X, and the denominator we have cosine multiplied by cosine squared, which is cosine cubed of X. Now from here, we're going to proceed with the U substitution. Let's suppose that U is equal to cosine of X. Let's differentiate you with respect to X, the divided by DX is the derivative of cosine of X. And it is going to be equal to negative sign of X. Now, if we multiply both sides by negative DX, we're going to get negative DU. Is equal to. Positive sign of X. Multiplied by DX. This is exactly what we have in the numerator sin x multiplied by DX, meaning now we can rewrite our integral in terms of U and DU. In the numerator, we have negative DU. We can also factor out the -1. And in the denominator we have cosine cubed of X, which is u cubed. This is now an elementary integral. We can integrate it by rewriting you. Cubed in the denominator as u to the power of -3 followed by the U. And now let's apply the power rule. We have negative. The integral of U to the power of -3 is going to be U to the power of -2 divided by -2. Plus a constant of integration C. We can now simplify, negative of negative gives us positive, so we get 1/2. U to the power of negative 2 + C, which is equal to 1 divided by 2 U 2 + C, right? And this is going to be equal to 1 divided by 2 multiplied by cosine squared of X plus C. Now, one divided by cosine is sun, so we got squared multiplied by 2, or basically 2 squared of X. Plus C instead of 2, we actually have 1/2, right, because we have 1/2 multiplied by 1 divided by cosine of X2. So that's 1/2 c 2 X + C, which is our final answer for this problem. Thank you for watching.

Use any method to evaluate the integrals in Exercises 65–70.∫ cot(x) / cos²(x) dx

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Use any method to evaluate the integrals in Exercises 65–70.
∫ cot(x) / cos²(x) dx