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11. ∫Tan³xSec⁴xdx ∫sec²xsec²xtan³xdx

∫(1 + tan²x)sec²xtan³xdx

u = gx , u = tanx , du = sec²x dx

<img src="https://tex.z-dn.net/?f=%20%5Cfrac%7Bdu%7D%7Bdx%7D%20%3Dsec%5E%7B2%7D%20x" />

<img src="https://tex.z-dn.net/?f=%20%5Cint%5Climits%281%2B%20u%5E%7B2%7D%20%29%20sec%5E%7B2%7D%20xu%5E%7B3%7D%20cos%5E%7B2%7D%20xdu" />

∫u⁵du + ∫u³du

tan⁶x tan⁴x - - - - - - - - - + - - - - - - - - 6 4

Rpta : tan⁶x tan⁴x - - - - - - - - - + - - - - - - - - + C 6 4

12.

∫<img src="https://tex.z-dn.net/?f=%20%5Cfrac%7B%20x%5E%7B2%7D%20%7D%7B%20%5Csqrt%7B4-%20x%5E%7B2%7D%20%7D%20%7Ddx%20" />

x = 2sin(u)

<img src="https://tex.z-dn.net/?f=%20%5Cfrac%7Bdx%7D%7Bdu%7D%20%3D2cos%28u%29" />

dx = 2cos(u) du

∫<img src="https://tex.z-dn.net/?f=%20%5Cfrac%7B8sin%5E%7B2%7D%28u%29cos%28u%29%20%7D%7B%20%5Csqrt%7B4-4sin%5E%7B2%7D%28u%29%20%7D%20%7D%20" />

<img src="https://tex.z-dn.net/?f=8.%20%5Cfrac%7B1%7D%7B2%7D%20.%20%5Cfrac%7B1%7D%7B2%7D%20%28%20%5Cint%5Climits1du-%20%5Cint%5Climitscos%282u%29du%29" />

<img src="https://tex.z-dn.net/?f=2%28arcsin%28x%2F2%29-1%2F2sin%282arcsin%28x%2F2%29%29%29" />

Rpta : <img src="https://tex.z-dn.net/?f=2%28arcsin%28x%2F2%29-1%2F2sin%282arcsin%28x%2F2%29%29%29%2BC" />

13.

∫<img src="https://tex.z-dn.net/?f=%20%5Cfrac%7B1%7D%7Bx%20%5Csqrt%7B9%2B%20x%5E%7B2%7D%20%7D%20%7D%20dx" />

u = <img src="https://tex.z-dn.net/?f=%20%5Csqrt%7B9%2B%20x%5E%7B2%7D%20%7D%20" /> du = x / u dx

∫[img = 10]

u = 3v du = 3dv

∫[img = 11]

[img = 12]

[img = 13]

Rpta : [img = 14] + C

14.

∫[img = 15]

u = x² + 9 du = 2xdx

∫[img = 16]

[img = 17]

u = x² + 9

1 / 2In / x² + 9 /

Rpta : 1 / 2In / x² + 9 / + c

15.

∫[img = 18]

u = x² du = 2xdx

∫[img = 19]

[img = 20]

v = [img = 21] dv = [img = 22]

[img = 23]

∫[img = 24]

[img = 25]

Rpta : [img = 26] + C

16.

∫[img = 27]

u = x² du = 2xdx

∫[img = 28]

v = 9u + 49 dv = 9du

[img = 29]

[img = 30]

Rpta : [img = 31] + C

17.

∫[img = 32]

∫[img = 33]

[img = 34]

2In / x - 4 / + 3In / x /

Rpta : 2In / x - 4 / + 3In / x / + C

18 .

[img = 35]

∫[img = 36]

∫[img = 37]

5In / x - 6 / - 4In / x + 2 /

Rpta : 5In / x - 6 / - 4In / x + 2 / + C

19.

∫[img = 38]

[img = 39]

∫[img = 40]

6(In / x - 1 / - [img = 41]) - ( - [img = 42])

[img = 43]

Rpta : [img = 44] + C

20.

∫[img = 45]

∫[img = 46]

u = (x + 1) du = 1dx

[img = 47]

[img = 48]

[img = 49]

Rpta : [img = 50] + C.