1-1-2-x-x-1-dx-2-0-2-2sinx-4cosx-sinx-cosx-dx-

Question Number 174341 by behi834171 last updated on 31/Jul/22

1 . \underset{1}{\int^{2}} \sqrt{x + \sqrt{x - 1}} d x = ?

2 . \underset{0}{\int^{\frac{π}{2}}} \frac{2 s i n x + 4 c o s x}{s i n x + c o s x} d x = ?

Commented by mnjuly1970 last updated on 30/Jul/22

2 . (3 π / 2)

Answered by MJS_new last updated on 30/Jul/22

1 .

\int \sqrt{x + \sqrt{x - 1}} d x =

[t = \sqrt{x - 1} + \frac{1}{2} \to d x = (2 t - 1) d t]

= \frac{1}{2} \int (2 t - 1) \sqrt{4 t^{2} + 3} d t =

[u = \frac{2 t + \sqrt{4 t^{2} + 3}}{\sqrt{3}} \to d t = \frac{\sqrt{4 t^{2} + 3}}{2 u} d u]

= \int (\frac{3 \sqrt{3}}{32} u^{2} - \frac{3}{16} u + \frac{3 \sqrt{3}}{32} - \frac{3}{8} u^{- 1} - \frac{3 \sqrt{3}}{32} u^{- 2} - \frac{3}{16} u^{- 3} - \frac{3 \sqrt{3}}{32} u^{- 4}) d u

the rest is easy

Commented by behi834171 last updated on 30/Jul/22

v e r y n i c e!

t h a n k y o u s o m u c h d e a r m a s t e r .

Answered by Frix last updated on 30/Jul/22

\underset{0}{\int^{π / 2}} \frac{2 \sin x + 4 \cos x}{\sin x + \cos x} d x =

= 3 \underset{0}{\int^{π / 2}} d x - \underset{0}{\int^{π / 2}} \frac{\sin x - \cos x}{\sin x + \cos x} d x

3 \underset{0}{\int^{π / 2}} d x = \frac{3 π}{2}

- \underset{0}{\int^{π / 2}} \frac{\sin x - \cos x}{\sin x + \cos x} d x =

= \underset{0}{\int^{π / 2}} \frac{d [\sin x + \cos x]}{\sin x + \cos x} =

= {[\ln ∣ \sin x + \cos x ∣]}_{0}^{π / 2} = 0

\Rightarrow answer is \frac{3 π}{2}

Commented by behi834171 last updated on 30/Jul/22

t h a n k y o u v e r y m u c h s i r .

y o u r a n s w e r i s t r u e .

Answered by behi834171 last updated on 31/Jul/22

l e t : x - 1 = t^{2} \Rightarrow d x = 2 t d t

I = \int 2 t \sqrt{t^{2} + 1 + t} d t = \int (2 t + 1 - 1) \sqrt{t^{2} + t + 1} d t =

= \int (2 t + 1) \sqrt{t^{2} + t + 1} d t - \int \sqrt{t^{2} + t + 1} d t = I_{1} + I_{2}

I_{1} = \frac{2}{3} {(t^{2} + t + 1)}^{\frac{3}{2}} + c o n s t .

I_{2} = \int \sqrt{{(t + \frac{1}{2})}^{2} + {(\frac{\sqrt{3}}{2})}^{2}} d t =

= \frac{2 t + 1}{4} \sqrt{t^{2} + t + 1} + \frac{3}{8} {s i n h}^{- 1} (\frac{2 t + 1}{\sqrt{3}}) + c o n s t .

\Rightarrow I = \frac{2}{3} {(x + \sqrt{x - 1})}^{\frac{3}{2}} + (\frac{2 \sqrt{x - 1} + 1}{4}) . \sqrt{x + \sqrt{x - 1}} +

+

\frac{3}{8} {s i n h}^{- 1} (\frac{2 \sqrt{x - 1} + 1}{\sqrt{3}}) + c o n s t .

[I_{1} = 2 \sqrt{3} - \frac{2}{3} = \frac{6 \sqrt{3} - 2}{3}

I_{2} = \frac{3 \sqrt{3} - 1}{4} + \frac{3}{10} = \frac{15 \sqrt{3} + 1}{20}

\Rightarrow I = I_{1} + I_{2} = \frac{165 \sqrt{3} - 37}{60}]

Commented by peter frank last updated on 01/Aug/22

thanks