Mathematics

Integrate $$\dfrac{\tan^4 \sqrt x+ \sec^2 \sqrt x}{\sqrt x}$$
The solution is $$\dfrac {2\tan ^3(\sqrt x)}{m}-2\tan \sqrt x+2\sqrt x+2\tan \sqrt x+C$$.Find m


ANSWER

3


SOLUTION

$$let\sqrt x=t\\then(\frac{1}{2\sqrt x})dx=dt\\\therefore I=2\int(tan^4t+sec^2t)dt\\=2\int[tan^2t(sec^2t-1)+sec^2t]dt\\=2\int tan^2t sec^2t dt-2\int tan^2t dt+2\int sec^2t dt \\=2(\frac{tan^3t}{3})-2\int(sec^2t-1)dt+2tant+c\\=(\frac{2}{3})tan^3t-2tant+2t+2tant+c \\= (\frac{2}{3})tan^3t-4tant +2t+c\\=(\frac{2}{3})tan^3t(\sqrt x)-2tan\sqrt x + 2tan\sqrt x +2\sqrt x+c\\\therefore m=3$$

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