Explanation
We have,
I=\int{\dfrac{dx}{\sqrt{9-25{{x}^{2}}}}}
I=\int{\dfrac{dx}{5\sqrt{\dfrac{9}{25}-{{x}^{2}}}}}
I=\dfrac{1}{5}\int{\dfrac{dx}{\sqrt{{{\left( \dfrac{3}{5} \right)}^{2}}-{{x}^{2}}}}}
We know that
\int{\dfrac{dx}{\sqrt{{{a}^{2}}-{{x}^{2}}}}}={{\sin }^{-1}}\left( \dfrac{x}{a} \right)+C
Therefore,
I=\dfrac{1}{5}{{\sin }^{-1}}\left( \dfrac{x}{\dfrac{3}{5}} \right)+C
I=\dfrac{1}{5}{{\sin }^{-1}}\left( \dfrac{5x}{3} \right)+C
Hence, this is the correct answer.
\displaystyle \int_{1}^{e^3}\dfrac{dx}{x\sqrt{1+ln x}}=\displaystyle \int_{1}^{e^3}\dfrac{dlnx}{\sqrt{1+ln x}}
\displaystyle \int_{1}^{e^3}\dfrac{dlnx}{\sqrt{1+ln x}}=2\sqrt{1+lnx}\displaystyle \displaystyle \vert_{1}^{e^3}
=2\sqrt{1+3}-2\sqrt{1}
=2\sqrt{4}-2=2(1)
=2
So, \displaystyle \int_{1}^{e^3}\dfrac{dx}{x\sqrt{1+ln x}}=2
\displaystyle \int_{0}^{\dfrac{\pi}{3}}\dfrac{\cos x }{3+4 \sin x}dx=\int_{0}^{\dfrac{\pi}{3}}\dfrac{d(\sin x)}{3+4 \sin x}
\displaystyle =\int_{0}^{\dfrac{\pi}{3}}\dfrac{d (\sin x) }{3+4 \sin x}=\dfrac{1}{4}\left[\log (3+4 \sin x)\right]_{0}^{\dfrac{\pi}{3}}
\displaystyle =\dfrac{1}{4}\left[\log \left[3+4\left(\sin \dfrac{\pi}{3}\right)\right]-\log (3+4 \sin (0))\right]
=\dfrac{1}{4}\left (\log\left( 3+\dfrac{4\sqrt{3}}{2}\right)-\log (3)\right)
=\dfrac{1}{4}(\log (3+2\sqrt{3})-\log 3)
=\dfrac{1}{4}\log \left ( \dfrac{3+2\sqrt{3}}{3} \right )
\therefore \displaystyle \int_{0}^{\dfrac{\pi}{3}}\dfrac{\cos x}{3+4 \sin x}dx=\dfrac{1}{4} \log \left(\dfrac{3+2\sqrt{3}}{3}\right)
\displaystyle \int_{0}^{\dfrac{\pi}{2}}\dfrac{cos x}{1+sin^2 x}dx=\displaystyle \int_{0}^{\dfrac{\pi}{2}}\dfrac{d sin x}{1+sin^2 x}
\displaystyle \int_{0}^{\dfrac{\pi}{2}}\dfrac{d sin x}{1+sin^2 x}=\left [( tan^{-1}(sin x)+ c\right ) \displaystyle ]_{0}^{\dfrac{\pi}{2}}\ (\int\dfrac{1}{1+x^2}=\tan^{-1}x)
=\left ( tan^{-1}(sin ( \dfrac{\pi}{2})+ c \right )-\left ( tan^{-1}(sin 0)+ c\right )
=tan^{-1}(1)
=\dfrac{\pi}{4}
We need to find the value of \displaystyle \int_{-1}^{3} \left[ \tan^{-1}\left ( \dfrac{x}{x^2+1} \right) + \tan^{-1}\left ( \dfrac{x^2+1}{x}\right ) \right] dx
We know \tan^{-1}p+\cot^{-1}p=\dfrac{\pi}{2}
\Rightarrow \tan^{-1}p+\tan^{-1}\dfrac{1}{p}=\dfrac{\pi}{2}
So, \tan^{-1}\left ( \dfrac{x}{x^2+1} \right )+\tan^{-1}\left ( \dfrac{x^2+1}{x} \right )=\dfrac{\pi}{2}
\displaystyle \int_{-1}^{3}\left [ \tan^{-1} \left ( \dfrac{x}{1+x^2}\right ) + \tan^{-1}\left ( \dfrac{x^2+1}{x}\right ) \right ]dx
=\displaystyle \int_{-1}^{3}\dfrac{\pi}{2}dx
=\dfrac{\pi}{2}\int_{-1}^{3}1\cdot dx
=\dfrac{\pi}{2}[3-(-1)]
=\dfrac{\pi}{2}\times 4=2\pi
\displaystyle \int_{\frac{1}{\sqrt{3}}}^{k}\dfrac{1}{1+x^2}dx=\dfrac{\pi}{6}
\displaystyle \int_{\frac{1}{\sqrt{3}}}^{k}\dfrac{1}{1+x^2}dx=tan^{-1}x\displaystyle \vert _{\frac{1}{\sqrt{3}}}^{k}
=[tan^{-1}k-tan^{\dfrac{-1}{\sqrt{3}}}]
=tan^{-1}k-tan^{-1}\dfrac{1}{\sqrt{3}}=\dfrac{\pi}{6}
tan^{-1}\dfrac{1}{\sqrt{3}}=\dfrac{\pi}{6} ---- equation (i)
tan^{-1}k=\dfrac{\pi}{6}+\dfrac{\pi}{6} ---- equation (ii)
tan^{-1}k=\dfrac{\pi}{3}
k=\sqrt{3}
\displaystyle I=\int_{0}^{\dfrac{\pi}{4}}\dfrac{\sin ^9 x}{\cos ^{11}x}dx=\int_{0}^{\dfrac{\pi}{4}}\dfrac{\sin ^9 x}{\cos ^{9}x} \sec ^2 dx
\displaystyle =\int_{0}^{\dfrac{\pi}{4}} \tan ^9x d(\tan x)
=\left[\dfrac{\tan ^{10}x}{10}+c \right]_{0}^{\dfrac{\pi}{4}}
=\left [ \left ( \dfrac{\tan ^{10}(\dfrac{\pi}{4})}{10}+c \right ) -\left ( \dfrac{\tan ^{10}(0)}{10} +c \right )\right ]
\tan \dfrac{\pi}{4}=1 ; \tan 0=0
\Rightarrow I=\dfrac{1}{10}-\dfrac{0}{10}=\dfrac{1}{10}
So, \displaystyle \int_{0}^{\dfrac{\pi}{4}}\dfrac{\sin ^9 x}{\cos ^{11}x}dx=\dfrac{1}{10}
We need to find value of \displaystyle \int_{1}^{2}\dfrac{1}{x\sqrt{x^2-1}}dxWe know that
\displaystyle \int \dfrac{1}{x\sqrt{x^2-1}}dx=\sec^{-1}x+c
\displaystyle \int_{1}^{2}\dfrac{1}{x\sqrt{x^2-1}}dx= \sec^{-1}2-\sec^{-1}1
\sec^{-1}2 =\dfrac{\pi}{3}
and \sec^{-1}1=0
So, \sec^{-1}2-\sec^{-1}1=\dfrac{\pi}{3}
\therefore \displaystyle \int_{1}^{2}\dfrac{1}{x\sqrt{x^2-1}}dx=\dfrac{\pi}{3}
\int_{0}^{\dfrac{\pi}{2}}\dfrac{cos x}{1+sin x}dx we know that \int_{0}^{\dfrac{\pi}{2}}\dfrac{cos x}{1+sin x}dx=\int_{0}^{\dfrac{\pi}{2}}\dfrac{d(sin x)}{1+sin x} \int_{0}^{\dfrac{\pi}{2}}\dfrac{d (sin x)}{1+sin x}=log (1+sin x)\int_{0}^{\dfrac{\pi}{2}} =log (1+sin \dfrac{\pi}{2}) - log (1+sin 0) =log (2) =log 2
\int_{0}^{\infty}xe^{-x^2}dx=\dfrac{1}{2}\int_{0}^{\infty}e^{-x^2}dx^2 =\dfrac{1}{2}\int_{0}^{\infty}e^{-x^2}dx^2=\dfrac{1}{2}\left [ -\dfrac{1}{e^{x^2}}+c \right ] \int_{0}^{\infty} =\dfrac{1}{2}\left [ (0+e)-(-1+e) \right ] =\dfrac{1}{2} \int_{0}^{\infty}xe^{-x^2}dx=\dfrac{1}{2}
\displaystyle \int_{0}^{1}\dfrac{x^2dx}{1+x^2}
\displaystyle \int_{0}^{1}1{dx}-\displaystyle \int_{0}^{1}\dfrac{1}{1+x^2}dx
=1-\left [ \tan^{-1}x \right ] \displaystyle \vert_{0}^{1}
=1 - \left [ \tan^{-1} 1 \right ]
=1 -\dfrac{\pi}{4}
Thus \displaystyle \int_{0}^{1}\dfrac{x^2}{1+x^2}dx=1-\dfrac{\pi}{4}
\int_{0}^{1}\dfrac{e^x}{1+e^{2x}}dx=\int_{0}^{1}\dfrac{de^x}{1+e^{2x}} =tan^{-1}(e^x)\int_{0}^{1} =tan^{-1}(e)-tan^{-1}(e^{\circ}) =tan^{-1}(e)-tan^{-1}(1) =tan^{-1}(e)-\dfrac{\pi}{4} So, \int_{0}^{1}\dfrac{e^x}{1+e^{2x}}dx=tan^{-1}e-\dfrac{\pi}{4}
\int_{1}^{2}\left ( \dfrac{1+x \log x}{x} \right ) e^x dx \int_{1}^{2}\left ( \dfrac{1}{x} + \log x \right ) e^xdx \displaystyle \int e^x [f(x)+f'(x) ] dx=e^x f(x) = \int_{1}^{2} \left ( e^x \log x+ c\right ) =e^2 \log 2-e \log 1 =e^2 \log 2-0 \int_{1}^{2}\left ( \dfrac{1+x \log x}{x} \right ) e^x dx=e^2 \log 2
\displaystyle \int_{\frac{\sqrt{2}}{3}}^{\frac{\sqrt{3}}{3}} \dfrac{dx}{2\sqrt{1-\dfrac{ax^2}{2}}}=\dfrac{1}{2}\displaystyle \int_{\frac{\sqrt{2}}{3}}^{\frac{\sqrt{3}}{3}} \dfrac{dx}{\sqrt{1-(\dfrac{3x}{2})^2}}
=\left[\dfrac{2}{3}\times \dfrac{1}{2} sin^{-1}\left(\dfrac{3x}{2}\right)\displaystyle \right]_{\frac{\sqrt{2}}{3}}^{\frac{\sqrt{3}}{3}}
=\dfrac{2}{3}\times \dfrac{1}{2} \left [ sin^{-1} \left ( \dfrac{3\sqrt{3}}{3(2)} \right ) -sin^{-1} \left ( \dfrac{3\sqrt{2}}{3\times 2} \right ) \right ]
=\dfrac{1}{3} \left [ sin^{-1} \left ( \dfrac{\sqrt{3}}{2} \right ) -sin \left ( \dfrac{1}{\sqrt{2}} \right ) \right ]
=\dfrac{1}{3} \left [ \dfrac{\pi}{3}-\dfrac{\pi}{4} \right ]
=\dfrac{1}{3} \left ( \dfrac{\pi}{12} \right )
=\dfrac{\pi}{36}
\displaystyle \int_{0}^{\dfrac{\pi}{2}} \dfrac{1}{sin x + cos x }dx
\displaystyle \int_{0}^{\dfrac{\pi}{2}} \dfrac{1+tan^2 \dfrac{x}{2}}{1-tan^2\dfrac{x}{2}+2 tan \dfrac{x}{2}} dx
\displaystyle \int_{0}^{\dfrac{\pi}{2}} \dfrac{2 d (tan \dfrac{x}{2})}{1-(tan\dfrac{x}{2}+1)^2}=2 \displaystyle \int_{0}^{\dfrac{\pi}{2}} \dfrac{d (tan \dfrac{x}{2})}{(\sqrt{2})^2-(tan \dfrac{x}{2}-1)^2}
=2\left [ \dfrac{1}{2\sqrt{2}} \right ] log \left | \dfrac{\sqrt{2}+tan\dfrac{x}{2}-1}{\sqrt{2}-(tan\dfrac{x}{2}-1)} \right | _{0}^{\dfrac{\pi}{2}}
=\dfrac{1}{\sqrt{2}} log \left ( \dfrac{\sqrt{2}+2}{\sqrt{2}-2} \right )
=log \left ( \dfrac{1+\sqrt{2}}{1-\sqrt{2}} \right )
=\sqrt{2}log (\sqrt{2}+1)
\displaystyle \int_{0}^{\dfrac{\pi}{2}}cos^5x\cdot sin2x dx=\displaystyle \int_{0}^{\dfrac{\pi}{2}}cos^5x\cdot 2 sinx cosx dx
=2\displaystyle \int_{0}^{\dfrac{\pi}{2}} cos^6 x sinx dx=-2\displaystyle \int_{0}^{\dfrac{\pi}{2}}cos^6 x\cdot d(cos x)
=-2\displaystyle \int_{0}^{\dfrac{\pi}{2}}cos^6 x dcos x=-2\left ( \dfrac{cos^7x}{7} \right )\displaystyle \vert_{0}^{\dfrac{\pi}{2}}
=-2\left [ \dfrac{cos^7 x}{7}\displaystyle \vert_{0}^{\frac{\pi}{2}} \right ]
=-2\left ( \dfrac{-1}{7} \right)=\dfrac{2}{7}
Thus \displaystyle \int_{0}^{\dfrac{\pi}{2}}cos^5 x\times sin 2x dx=\dfrac{2}{7}
Consider, I=\displaystyle \int_{0}^{\dfrac{\pi}{4}}\sqrt{\dfrac{1-sin 2 x}{1+sin 2 x}}\cdot dx
\displaystyle I= \int_{0}^{\dfrac{\pi}{4}}\sqrt{\dfrac{(sinx - cos x)^2}{(sin x + cos x)^2}}\cdot dx
\displaystyle I=\int_{0}^{\dfrac{\pi}{4}}{\dfrac{(sinx - cos x)}{sin x + cos x}}\cdot dx
\displaystyle I=-\int_{0}^{\dfrac{\pi}{4}}{\dfrac{cos x- sin x}{sin x + cos x}}dx
I=-\left[log |sin x + cos x|\right]_0^{\frac{\pi}{4}}
I=-\left[log (sin {\frac{\pi}{4}} + cos {\frac{\pi}{4}}) -\log (\sin 0+\cos 0))\right]
I=-\left[log \left (\dfrac{1}{\sqrt2}+\dfrac{1}{\sqrt 2}\right)-\log 1\right]
I=-log \left ( \dfrac{1}{\sqrt 2} +\dfrac{1}{\sqrt 2} \right ) =-log \sqrt{2}
So, \displaystyle \int_{0}^{\dfrac{\pi}{4}}\sqrt{\dfrac{1- sin 2 x}{1+sin 2 x}}x=-log (\sqrt{2})
\displaystyle \int_{0}^{1}\dfrac{dx}{\sqrt{1-x^2}}
\displaystyle \int \dfrac{dx}{\sqrt{1-x^2}}=\sin^{-1}x+c
\displaystyle \int_{0}^{1}\dfrac{dx}{\sqrt{1-x^2}}=\left[(\sin^{-1}x+c) \right]_{0}^{1}
=(\sin^{-1}1+c)-(\sin^{-1}0+c)
=\dfrac{\pi}{2}
\displaystyle \int_{0}^{1}\dfrac{dx}{\sqrt{1-x^2}}=\dfrac{\pi}{2}
\displaystyle \int_{0}^{1}\dfrac{(sin^{-1}x)^2}{\sqrt{1-x^2}} \cdot dx
\displaystyle \int \dfrac{(sin^{-1}x)^2}{\sqrt{1-x^2}} \cdot dx=\displaystyle \int (sin^{-1}x)d(sin^{-1}x)
=\dfrac{(sin^{-1}x)^3}{3}+c
\displaystyle \int_{0}^{1}\dfrac{(sin^{-1}x)^2}{\sqrt{1-x^2}} \cdot dx=\left[\dfrac{(sin^{-1}x)^3}{3}+c\right]_{0}^{1}
=\left ( \dfrac{(sin^{-1}1)^3}{3}+c\right )-\left ( \dfrac{(sin^{-1}0)^3}{3}+c\right )
=\dfrac{1}{3}\left ( \dfrac{\pi}{2} \right )^3
=\dfrac{\pi^3}{24}
\displaystyle \int_{\frac{1}{2}}^{1}\dfrac{1}{\sqrt{1-x^2}}dx
We know that \displaystyle \int \frac{1}{\sqrt{1-x^2}}dx=(\sin^{-1}x+c)
\displaystyle \int_{\frac{1}{2}}^{1}\dfrac{1}{\sqrt{1-x^2}}dx=\left[(\sin^{-1}x+c)\displaystyle \right]_{\frac{1}{2}}^{1}
=(sin^{-1}1+c)-(\sin{\dfrac{1}{2}}+c)
=\sin^{-1}1-\sin^{-1}\dfrac{1}{2}
=\dfrac{\pi}{2}-\dfrac{\pi}{6}
=\dfrac{\pi}{3}
\displaystyle \int_{\frac{1}{2}}^{1}\dfrac{1}{\sqrt{1-x^2}}dx=\dfrac{\pi}{3}
\int_{0}^{1}\sqrt{1-x^2}\cdot dx \int \sqrt{1-x^2}dx=\dfrac{x}{2}\sqrt{a^2-x^2}+\dfrac{a^2}{2} sin^{-1}\left ( \dfrac{x}{a} \right )+ c \int_{0}^{1}\sqrt{1-x^2}\cdot dx= \left ( \dfrac{1}{2} \sqrt{1^2-x^2}+\dfrac{a^2}{2}sin^{-1} \left ( \dfrac{x}{a} \right )+c \right )\int_{0}^{1} =\left ( \dfrac{1}{2} (0)+ \dfrac{1}{2} sin^{-1}(1)+c \right ) - (0+0+c) =\dfrac{1}{2}\dfrac{\pi}{2} =\dfrac{\pi}{4} \int_{0}^{1}\sqrt{1-x^2}dx=\dfrac{\pi}{4}
\int_{0}^{a}\dfrac{1}{a^2+x^2}dx
\int \dfrac{1}{a^2+x^2}dx=\dfrac{1}{a^2}
\int \dfrac{d(\dfrac{x}{a})}{1+(\dfrac{x}{a})^2}=\dfrac{1}{a}\int \dfrac{d(\dfrac{x}{a})}{1+(\dfrac{x}{a})^2}
=\dfrac{1}{a}tan^{-1}(\dfrac{x}{a})+c
\int_{0}^{a}\dfrac{1}{a^2+x^2}dx=\left [ \dfrac{1}{a} tan^{-1}\left ( \dfrac{x}{a} \right ) +c \right ] \int_{0}^{a}
=\left [ \dfrac{1}{a}tan^{-1}\left ( \dfrac{a}{a} \right ) +c \right ] -(0+c) =\dfrac{1}{a}tan^{-1}(1)
=\dfrac{1}{a}\cdot \dfrac{\pi}{4}
=\dfrac{\pi}{4a}
\displaystyle \int_{a}^{\frac{a}{2}}\dfrac{1}{\sqrt{a^2-x^2}}\cdot dx
\displaystyle \int \dfrac{1}{\sqrt{a^2-x^2}}\cdot dx=\dfrac{1}{a}\displaystyle \int \dfrac{1}{\sqrt{1-\left ( \dfrac{x}{a} \right )^2}}\cdot dx
=\displaystyle \int \dfrac{d\left ( \dfrac{x}{a} \right )}{\sqrt{1-\left ( \dfrac{x}{a} \right )^2}}
=\sin^{-1}(\dfrac{x}{a})+c
\displaystyle \int_{\frac{a}{2}}^{a}\dfrac{1}{\sqrt{a^2-x^2}}\cdot =\left [ \sin^{-1}(\dfrac{x}{a})+c \right ] _{\frac{a}{2}}^{a}
=(\sin^{-1}(\dfrac{a}{a})+c)-(\sin^{-1}(\dfrac{\frac{a}{2}}{a})+c)
=\left [ (\sin (1)+c ) -(\sin^{-1}(\dfrac{1}{2})+c) \right ]
\displaystyle \int_{0}^{1}\dfrac{x}{a+x^2}dx
\displaystyle \int \dfrac{x}{1+x^2}dx=\dfrac{1}{2}\displaystyle \int \dfrac{dx^2}{1+x^2}=\dfrac{1}{2} \log (1+x^2)+c
=\displaystyle \int_{0}^{1}\dfrac{x}{1+x^2}dx=\left [ \dfrac{1}{2} \log (1+x^2)+c \right ]_{0}^{1}
=\left ( \dfrac{1}{2} \log (2)+c \right ) - \left ( \dfrac{1}{2} \log (1)+c\right )
=\dfrac{1}{2} \log 2
\displaystyle \int_{0}^{1}\dfrac{x}{1+x^2}dx=\dfrac{1}{2} \log 2
Let \displaystyle I=\int_{0}^{1}\dfrac{x^3}{1+x^8} dx
Now, \displaystyle \int \dfrac{x^3}{1+x^8 }dx =\dfrac{1}{4}\int \dfrac{d(x^4)}{1+x^8}=\dfrac{1}{4}\tan^{-1}(x^4)+c
I=\displaystyle \int_{0}^{1}\dfrac{x^3}{1+x^8}dx=\left(\dfrac{1}{4}\tan^{-1}(x^4)+c\right)_{0}^{1}
=\left [\dfrac{1}{4} \tan^{-1}(1)+c\right ]-\left [\dfrac{1}{4} \tan^{-1}(0)+c\right ]
=\dfrac{1}{4} \tan^{-1}(1)
=\dfrac{1}{4} \cdot \dfrac{\pi}{4}=\dfrac{\pi}{16}
Hence, \displaystyle \int_{0}^{1}\dfrac{x^3}{1+x^8}dx =\dfrac{\pi}{16}
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