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Evaluate the integral.

$ \displaystyle \int \tan^4 x \sec^6 x dx $

$$\frac{1}{9} \tan ^{9} x+\frac{2}{7} \tan ^{7} x+\frac{1}{5} \tan ^{5} x+C$$

Integration Techniques

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this problem is from Chapter seven section to problem number twenty five in the book Calculus Early. Transcendental lt's a Tradition by James Door Here we have an indefinite a roll of tangent to the fourth power of EXT. Time sequence of the Six Power of X. So the first thing we can do here is rewrite the sea cans of the six power. So let's write as seeking Square X square. So that c cancer, The Fourth times seeking Square of X, the ex. And now we can use one of our protagonist identities to rewrite this Sikh and squared of X. And we have integral ten to the fourth power, vex parentheses and then seek and squared. Using our battalion identity becomes tan squared X plus one and that's also being squared time seeking and square eggs. The ex. Now we see we can apply a u substitution. Let's take you to be Tana Becks. So then do you become sick and square? Vicks the ex? So here we see. We have to you ten to the fourth is simply you do the fourth and then we have use flair plus one in the parentheses. And that's also being square. So our inner girl becomes after applying the substitution you to the fourth power you squared plus one square, do you Before we integrate, let's go ahead and simplify. This is much as we can. We have you to the fourth. So let's evaluate this square. We have you to the fourth and the parentheses. Plus two, you squared plus one and let's go ahead and distribute issue to the fourth through the apprentices. So have you ate Plus, to you to the six Clinton plus you know, the forth to you. And now we can apply the power rules. Evaluate each of these three intervals. So doing So we have you threw the knife power over nine plus to you to the seven number seven. Plus, you know, the fifth power over five plus our constant of immigration. See, And finally we come back to our original U substitution so that we can back substitute you with Tanox so that we have ten to the ninth Power Becks Overnight plus two, ten, seven. Power affects over seven, plus stance of the fifth power of X over five and plus our constancy. And there's our answer