2–74. Integration techniques Use the methods introduced in Sec...

Question

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.

6. ∫ (2 − sin 2θ)/cos² 2θ dθ

Accepted Answer

Welcome back, everyone. Evaluate the integral integral of 3 + sin of 4 X, all divided by cosine squared of 4 X and multiplied by DX. For this problem, let's rewrite the integral as a sum of two integrals. So first of all, we have 3 divided by cosine squared of 4 X D X plus our next integral is sin of 4 X. Divided by cosine squared of 4 X multiplied by DX. So for the first integral, let's factor out 3, so we get 3 integral, one divided by cosine squared of 4 XDX. And for the second integral, let's rewrite cosine squared as the product of two cosines. So we get sine of 4 X divided by cosine of 4 X multiplied by 1 divided by cosine of 4 X, and we're going to add the X in the numerator. So now what we can see is that we're evaluating the integral of one divided by cosine squared of for X, right? And this can be written as the integral of 2 squared of 4 XDX. And for the second integral, we can rewrite it as tangent of 4 X because we have a ratio of sine and cosine. Multiplied by sea and. Of 4 XD X, right, because we have 1 divided by cosine of 4 X, which is basically sequent of 4 X. And now let's introduce U substitution. Let's suppose that u equals 4 X, then the derivative DU divided by D X is 4, and this means that the X is equal to DU divided by 4, or basically 1/4 DU. And we're going to apply this substitution for each integral. So let's label our sum of integrals as I, and then we can rewrite I in terms of you. So we get 3 integrals can squared of 4 XDX essentially becomes squared of you. Multiplied by the X, which is 14th DU, so we're going to write the U and factor out 14th. So in front of the integral we get 3 divided by 4. For the next integral. We are integrating the ancient of uses, 2 of you. Multiplied by the X so that bed u divided by 4 once again we can factor out the constant and now we are ready to integrate. So we got 34. Now, the integral of second squared UDU is going to be equal to tangent of U because it is a basic integral, right? And then for the next integral, we have 1/4th we're integrating tangent to use sequent U, which is equal to sequent of U. And we're adding a constant of integration C and finally let's substitute back. Our expression in terms of X, so we get 3/4 tangent of 4 X plus 1/4 of 4 X plus a constant of integration C. That'll be our final answer and thank you for watching.

2–74. Integration techniques Use the methods introduced in Sections 8.1 through 8.5 to evaluate the following integrals.
6. ∫ (2 − sin 2θ)/cos² 2θ dθ

Key Concepts

Integration Techniques

Trigonometric Identities

Substitution Method

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