DB45
1) ∫ tan x sec^5 x + sin^(-1/4) (8x) cos (8x) dx = 1/5 sec^5 x - 32/3 sin^3/4 8x + c = 1/5 sec^5 x - 32/3 ⁴√(sin³ (8x)) + c
2) = 4 ∫ ( 2 sin 9x cos 3x ) +3∫ (-2 sin 9x sin 3x) = 4 ∫ sin 12x + sin 6x + 3 ∫cos 12x - cos 6x = 4 [-1/12 cos 12x - 1/6 cos 6x + 3[ 1/2 sin 12x - 1/6 sin 6x + c = - 1/3 cos 12x - 2/3 cos 6x + 1/4 sin 12 x - 1/2 sin 6x + c
![int( \frac{sin x}{cos^{6}x }+ \frac{cos8x}{ \sqrt[4]{sin8x} } ) dx](https://tex.z-dn.net/?f=int%28+%5Cfrac%7Bsin+x%7D%7Bcos%5E%7B6%7Dx+%7D%2B+%5Cfrac%7Bcos8x%7D%7B+%5Csqrt%5B4%5D%7Bsin8x%7D+%7D+%29+dx "int( \frac{sin x}{cos^{6}x }+ \frac{cos8x}{ \sqrt[4]{sin8x} } ) dx")
∫ tan x sec^5 x + sin^(-1/4) (8x) cos (8x) dx
= 1/5 sec^5 x - 32/3 sin^3/4 8x + c
= 1/5 sec^5 x - 32/3 ⁴√(sin³ (8x)) + c
2)
= 4 ∫ ( 2 sin 9x cos 3x ) +3∫ (-2 sin 9x sin 3x)
= 4 ∫ sin 12x + sin 6x + 3 ∫cos 12x - cos 6x
= 4 [-1/12 cos 12x - 1/6 cos 6x + 3[ 1/2 sin 12x - 1/6 sin 6x + c
= - 1/3 cos 12x - 2/3 cos 6x + 1/4 sin 12 x - 1/2 sin 6x + c