Integrals

log? (18x-x²-77)>0 ⇒ 18x-x²-77>1 ⇒ x²-18x+78<0. Roots are 93.
log? (.)>0 ⇒ log? (.)>1 ⇒ 18x-x²-77>3 ⇒ x²-18x+80<0 (x-8) (x-10)<0.
8Domain is (8,10). a=8, b=10.
I = ∫? ¹? sin³x/ (sin³x+sin³ (18-x)dx. Using King's property.
I = ∫? ¹? sin³ (18-x)/ (sin³ (18-x)+sin³x)dx.
2I = ∫? ¹? dx = 2. I=1.

I = ∫? π/? π/? dx/ (1+e^ (xcosx) (sin? x+cos? x). Using ∫? f (x)dx = ∫? f (a+b-x)dx. a+b=0.
I = ∫? π/? π/? dx/ (1+e? ) (sin? x+cos? x) = ∫? π/? π/? e? dx/ (e? +1) (sin? x+cos? x).
2I = ∫? π/? π/? dx/ (sin? x+cos? x) = 2∫? π/? dx/ (sin? x+cos? x).
I = ∫? π/? sec? xdx/ (tan? x+1). Let t=tanx.
I = ∫? ¹ (t²+1)dt/ (t? +1) = ∫? ¹ (1+1/t²)dt/ (t²-√2t+1) (t²+√2t+1). No, this is hard.
I = ∫? ¹ (1+1/t²)dt/ (t-1/t)²+2). Let u=t-1/t. I = ∫ du/ (u²+2) = (1/√2)tan? ¹ (u/√2).
= π/ (2√2).

Given $l_{m,n}={\displaystyle {\int }_0^1{x}^{m-1}{\left(1-x\right)}^{n-1}dx............\left(i\right)}$  

put 1 - x = $t\{\begin{array}{l}x=0,t=1\\ x=1,t=0\end{array}$

dx = -dt

From (i) $l_{m,n}={\displaystyle {\int }_1^0{\left(1-t\right)}^{m-1}.t^{n-1}\left(-dt\right)}$

$l_{m,n}={\displaystyle {\int }_0^1{t}^{n-1}{\left(1-t\right)}^{m-1}dt={\displaystyle {\int }_0^1{x}^{n-1}{\left(1-x\right)}^{m-1}dx..........\left(ii\right)}}$    

$Putx=\frac{1}{1+y}in\left(i\right)thendx=-\frac{1}{{\left(1+y\right)}^2}dy$                          

(i)  $l_{m,n}={\displaystyle {\int }_{\infty }^0\frac{1}{{\left(1+y\right)}^{m-1}}}{\left(1-\frac{1}{1+y}\right)}^{n-1}\left(\frac{-dy}{{\left(1+y\right)}^2}\right)={\displaystyle {\int }_0^{\infty }\frac{y^{n-1}}{{\left(1+y\right)}^{m+n}}dy}..........\left(iii\right)$

Similarly by (ii) $l_{m,n}={\displaystyle {\int }_0^{\infty }\frac{y^{m-1}}{{\left(y+1\right)}^{m+n}}dy..........\left(iv\right)}$

Adding (iii) & (iv) $2l_{m,n}={\displaystyle {\int }_0^{\infty }\frac{y^{n-1}+y^{m+1}}{{\left(y+1\right)}^{m+n}}}$

Putting  $\frac{1}{z}\{\begin{array}{l}y=1,z=1\\ y=\infty ,z=0\end{array}$ $\Rightarrow dy=\frac{-1}{z^2}dz$  

Hence $l_{m,n}={\displaystyle {\int }_0^1\frac{x^{m-1}+x^{n-1}}{{\left(1+x\right)}^{m+n}}}$ dx = α lm, n

=> α = 1

${\displaystyle \underset{0}{\overset{\pi }{\int }}\left|sin2x\right|dx}$

$=2{\displaystyle \underset{0}{\overset{\pi /2}{\int }}sin2xdx}$ =2 ${\left[\frac{-cos2x}{2}\right]}_0^{\pi /2}$ = $2\left(\frac{1}{2}-\left(-\frac{1}{2}\right)\right)=2$

Information missing. The question was droppedby NTA.

 $y=\left|\left|x-1\right|-2\right|$

Area bounded region = $\frac{1}{2}*4*2=4$

$\underset{h\to 0}{Lt}\frac{2*2\left[\frac{\sqrt[]{3}}{2}sin\left(\frac{\pi }{6}+h\right)-\frac{1}{2}cos\left(\frac{\pi }{6}+h\right)\right]}{2\sqrt[]{3}h\left[\frac{\sqrt[]{3}}{2}cosh-\frac{1}{2}sinh\right]}$

$=\underset{h\to 0}{Lt}4\frac{sin\left(\frac{\pi }{6}+h-\frac{\pi }{6}\right)}{2\sqrt[]{3}hsin\left(\frac{\pi }{3}-h\right)}=\frac{2}{\sqrt[]{3}}*1*\frac{1}{\frac{\sqrt[]{3}}{2}}=\frac{4}{3}$

 ${\displaystyle \sum _{n=1}^{100}{\displaystyle \underset{n-1}{\overset{n}{\int }}{e}^{x-\left[x\right]}dx={\displaystyle \sum _{n=1}^{100}{\displaystyle \underset{n-1}{\overset{n}{\int }}{e}^{x-\left[x\right]}dx}}}}$

${\displaystyle \underset{0}{\overset{1}{\int }}{e}^{x-\left[x\right]}dx+{\displaystyle \underset{1}{\overset{2}{\int }}{e}^{x-\left[x\right]}dx}}+{\displaystyle \underset{2}{\overset{3}{\int }}{e}^{x-\left[x\right]}dx+.....+{\displaystyle \underset{99}{\overset{100}{\int }}{e}^{x-\left[x\right]}dx}}$

$=e-1+\frac{1}{e}\left(e^2-e\right)+\frac{1}{e^2}\left(e^3-e^2\right)+......+\frac{1}{e^{99}}\left(e^{100}-e^{99}\right)$

$=e-1+\left(e-1\right)+\left(e-1\right)+......+\left(e-1\right),100times$

$=100\left(e-1\right)$

2nd method

${\displaystyle \sum _{n=1}^{100}{\displaystyle \underset{n-1}{\overset{n}{\int }}{e}^{x-\left[x\right]}dx={\displaystyle \sum _{n=1}^{100}{\displaystyle \underset{n-1}{\overset{n}{\int }}{e}^{\left\{x\right\}}dx=}}{\displaystyle \sum _{n=1}^{100}\left({\displaystyle \underset{0}{\overset{1}{\int }}{e}^{x}dx}\right)}}}={\displaystyle \sum _{n=1}^{100}\left(e-1\right)}=100\left(e-1\right)$

 $l={\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{co{s}^{2}x}{1+3^x}dx..........\left(i\right)}$

Using properties, ${\displaystyle \underset{a}{\overset{b}{\int }}f\left(x\right)dx={\displaystyle \underset{a}{\overset{b}{\int }}f\left(a+b-x\right)dx}}$

$l={\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{co{s}^{2}x}{1+3^{-x}}dx}={\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{3^{x}co{s}^{2}x}{1+3^x}dx.......\left(ii\right)}$

Adding (i) and (ii), we get

$2l={\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}\frac{\left(1+3^x\right)co{s}^{2}x}{1+3^x}}dx={\displaystyle \underset{-\pi /2}{\overset{\pi /2}{\int }}co{s}^{2}xdx=2}{\displaystyle \underset{0}{\overset{\pi /2}{\int }}co{s}^{2}xdx}$

$\therefore l=\frac{\pi }{4}$

$\underset{x\to 0}{lim}\frac{{\displaystyle \underset{0}{\overset{x^2}{\int }}\left(sin\sqrt[]{t}\right)dt}}{x^3}$     $\left(\frac{0}{0}\right)$  applying L' Hospital rule

$=\underset{x\to 0}{lim}\frac{sin\left(\sqrt[]{x^2}\right).2x}{3x^2}$

$=\frac{2}{3},forx>0$

Area of shaded region

$2{\displaystyle \underset{0}{\overset{\sqrt[]{3}}{\int }}\left(2x^2+9-5x^2\right)dx}$

$=2{\displaystyle \underset{0}{\overset{\sqrt[]{3}}{\int }}\left(9-3x^2\right)dx}$

${\displaystyle \underset{0}{\overset{\sqrt[]{3}}{\int }}\left(3-x^2\right)dx=6{\left[3x-\frac{x^3}{3}\right]}_0^{\sqrt[]{3}}=6\left[\frac{9\sqrt[]{3}-3\sqrt[]{3}}{3}\right]}=12\sqrt[]{3}$