7–84. Evaluate the following integrals.
11. ∫ from 0 to ...

Question

7–84. Evaluate the following integrals.

11. ∫ from 0 to π/4 (sec x – cos x)² dx

Accepted Answer

Hello. In this video, we are going to be evaluating the following integral. We are given the integral from 0 to pi divided by 3 of the quantity seeking X plus sine X squared. Now, in order to approach this problem, let's first algebraically simplify the integral. We can simplify the integral by squaring seekin X plus sine of X squared, and that's going to allow us to rewrite the integral to be the integral from 0 to pi divided by 3 of skin squared. Of X plus 2 multiplied by seeking of X. Multiplied by sin of X. Plus, sine squared of XDX. Now, see X multiplied by sin of X will simplify to be tangent of X. So this middle term will further simplify to be tangent of X. Now, it is a little difficult to integrate these three terms together. So instead, let's go ahead and separate this integral into a sum of 3 definite integrals. That is going to be the integral from 0 to pi divided by 3 of sequent squared X. Plus 2 multiplied by 0 to pi divided by 3. So just to note, I remove this 2 and put it in front of the integral, since it's a constant of tangent of X. Plus the definite integral from 0 to pi divided by 3 of sine squared of X, and all this will be integrated with respect to D X. Let's go ahead and evaluate each of these integrals independently. So we'll label the first integral 1, the second integral 2, and the third integral 3. So, starting with the first integral, again, we have the integral from 0 to pi divided by 3 of sequin squared X. Now, skin squared is the derivative of tangent, so the integral of skin squared is going to be tangent. And this is going to be evaluated from 0 to pi divided by 3. This is going to give us tangent of pi divided by 3. Minus tangent of 0, and this is going to simplify to give us the square root of 3. We will take this value and send it to the site for now. Now let's approach the 2nd integral. So for the second integral, we have 2 multiplied by the integral from 0 to pi divided by 3 of tangent of XDX. Now, tangent of X isn't easy to integrate. In fact, we'll need to use a special identity in order to integrate tangent of X, but the special integral of tangent of X is going to be the natural logarithm of cosine of X. This is the antiderivative of tangent of X. Now keep in mind that we have this two coefficient in front of the integral, so we have to multiply 2 to this natural logarithm function. Furthermore, this is going to be evaluated from 0 to pi divided by 3. So by plugging in these values, we will have 2 multiplied by the natural logarithm of cosine of pi divided by 3. Oh, there's one thing I forgot about the natural logarithm function, and it's the fact that it's also negative. So, when we plug in pi divided by 3, we get -2 multiplied by the natural logarithm of cosine pi divided by 3. Then we'll subtract subtract -2, which gives us 2 multiplied by the natural logarithm of cosine of 0. And by simplifying this, we are going to be left with positive 2 multiplied by the natural logarithm of 2. This is going to be the value of the 2nd integral, and we'll set this to the site for now. And finally, we will now evaluate the 3rd integral. The third integral was the integral from 0 to pi divided by 3 of sin squared of X. Now, in order to approach this integral, we are going to have to use the power reduction formula. So, we can rewrite sine squared X to be the integral from 0 to pi divided by 3. Of 1/2 multiplied by 1 minus cosine of 2 X. No This is going to evaluate to leave us with one half multiplied by X. Multiplied or minus 1/2 of 2 X. And this will be evaluated from 0 to pi divided by 3. Furthermore, if we were to distribute this one half, we end up with X divided by 2 minus 1/4 of 2 X. And now we will need to evaluate this from 0 to pi divided by 3. Once we plug in these values, this is going to simplify to be pi divided by 6 minus the square root of 3 divided by 8, and this is going to be the value of the third integral. So, now if we plug in these values into the original integral, we can rewrite the original integral to be the square root of 3. Plus 2 multiplied by the natural logarithm of 2, plus pi divided by 6 minus the square root of 3 divided by 8. And by combining like terms together, we can simplify this to be 2 multiplied by the natural logarithm of 2, plus 21 multiplied by the square root of 3, plus 4 pi divided by 24, and this is going to be the final value of the given integral. And with that being said, the final solution to this problem is going to be C. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

7–84. Evaluate the following integrals.
11. ∫ from 0 to π/4 (sec x – cos x)² dx

Key Concepts

Integration

Trigonometric Functions

Definite Integrals

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