Problem 50 Hard Difficulty

If $ \displaystyle \int_0^{\frac{\pi}{4}} \tan^6 x \sec x dx = I $, express the value of $ \displaystyle \int_0^{\frac{\pi}{4}} \tan^8 x \sec x dx $ in terms of $ I $.

Answer

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Video Transcript

here we'd like to express the integral of attention to the eighth power time seek in in terms of the integral of tangents, of the six power of times he can't So here the first thing we could do is take this integral of ten eight times. He can't and just rewrite it by pulling that one factor of ten. So instead of Tanay, we have dance of the seven times ten then for this newer and a girl on the right side we could use integration, my parts. So let's take you to be a chance of the seventh so that by the chain rule we know that do you is seventeen to the sixth Power time's derivative of tension sequence where and then DX and we're left over with Devi to be Tim times he can you have a ten x seek an ex d x and we know the integral of this is vey wishes. Seek an ex So let's go ahead and apply our formula. So by integration, my parts we have you times feet. So the chance of the seventh time seeking and this is our UV and we have her end points zero in power before minus the integral part before of video. So it's going to pull out that seven. Then we have a tangents of the six power seek and square time seeking. So this is the minus minus in a rural need to you. So here we're using the integration by parts one. So a few things we can do here one the first thing we could do is evaluate this expression on the left by plugging in the end points. The other thing we can do is take this. C can't square and rewrite. This is tan squared plus one using your protagonist identities. So let's go ahead. And here I was, plugging the end point. First attention of power before is one. So we just have a once in the seventh and then seeking a power for a square, too. So this is when you plug in power before and when you plug in zero for X tangent zero zero. So we just attract nothing. So that's the first term. Now for the next term, let's go ahead and simplify this. So have I seven and the girls they are apart before. So after I multiply attention to the six power through the parentheses. We have tangent to the power times. He can't Great. And then we also have quite and split this up into two and minerals. We have negative seven and then we have ten to the six times, one times again and we can notice that this last and the girl over here is the same as the integral I that was given to us over here. So we we see already how we can express our general that we're starting with Wei appear on the left in terms of I but we have more simplification to do but that that this is where I was coming from. I also noticed that this circle this green terrible here is the same as the interval that we started with. So that suggests that we should push this negative seven to the other side to get a positive eight. No. So doing so. We have eight and a girl's ear apart before tangents to Dave equals one times swear too. So we have suero too, minus seven I and then dividing both sides by eight. We got our final answer. So we have square to over eight after dividing by eight. And then we have seven I over eight, and there's really no need to add a constancy of integration here because you could. It's already taken into account. By this. I turn because I is an animal so you could think of. I has already has the constant of integration so we could stop right here, and that's your final answer.