Evaluate the integrals in Exercises 33–52.
∫ sec⁶(x) dx

Question

Evaluate the integrals in Exercises 33–52.

∫ sec⁶(x) dx

Accepted Answer

Welcome back, everyone. Evaluate the integral of sequence to the power of 4 of XDX. For this problem, let's begin by remembering that squared of X can be written as 1 + tangent squared of x. This is a trigonometric identity. So if we consider sequence to the power of 4 of X, we can write it as squad of X multiplied by squad of X. Which is equal to 1 + tent squad of X multiplied by 2 squared X, right? And now what if we use a U substitution? Let's suppose that U is equal to tangent X. Then the derivative of u with respect to X is the derivative of tangent of X, which is c squared X, right? So if we multiply both sides by DX we can now show that DU is equal to square x multiplied by DX and therefore we can now rewrite our integral as sequence to the power of 4 of XDX. Equals integral of, we already know squad of X multiplied by squad of X, right? Multiplied by DX. Which is equal to the integral of 1 plus tangent squared of x multiplied by squared XD X. Now notice that tangent X is our U and squared XDX is our DU. So now the integral becomes. Integral of a 1 plus U squared d u. Which is an elementary integral. The integral of 1 is U, and the integral of U squared is u cubed divided by 3. Using the power rule. We're going to add the constant of integration C, and our final step is to replace you with tangent. So we get tangent of X. Plus Tangent cubed of X divided by 3. Plus a constant of integration C. That's our final answer. Let's label it and thank you for watching.

Evaluate the integrals in Exercises 33–52.∫ sec⁶(x) dx

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Evaluate the integrals in Exercises 33–52.
∫ sec⁶(x) dx