OpenStudy (arindameducationusc):

Examples of integration part1

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{Examples~of~integration}$$ check the below tutorial by @hartnn its just awesome.... http://openstudy.com/study#/updates/50960518e4b0d0275a3ccfba

3 years ago
OpenStudy (arindameducationusc):

referred with this tutorial I will be giving examples. All $$\Large\color{blue}{Serial~no.}$$ are according to the above tutorial. just see the above tutorial, check the serial number and you can see the example of the formula.

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{For~1.}$$ example is already in the tutorial. (check it)

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{2.}$$ $\int\limits_{}^{}\sin (5x+7)= -\frac{ 1 }{ 5 }\cos(5x+7) +C$ C is the arbitrary constant of integration.

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{3.}$$ $\int\limits_{}^{} e ^{xlog _{e}a} dx, ~a>0, ~a \neq1$ $$\Large\color{blue}{Solution(Soln)}$$ $\int\limits_{}^{}e ^{\log _{e}a ^{x}}dx$ and now using property of log we know $e ^{\log _{e}}=1$ so we get, $\int\limits_{}^{}a ^{x}dx$ $=\frac{ a ^{x} }{ \log _{e}a }+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{Q}$$ $\int\limits_{}^{} 10^{x}dx$ $$\Large\color{blue}{Soln}$$ $=\frac{ 10^{x} }{ \log10 }+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{Q.}$$ $\int\limits_{}^{}e ^{x}a ^{x}dx$ $$\Large\color{blue}{Soln}$$ $=\int\limits_{}^{}(ae)^{x}dx$ $=\frac{ (ae)^{x} }{ \log(ae) } + C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{4.}$$ $$\Large\color{red}{Q}$$ $I=\int\limits_{}^{}\frac{ x^{3}+5x^{2}+4x+1 }{ x^{2} } dx$ $$\Large\color{blue}{Soln}$$ =$\int\limits_{}^{}(x+5+\frac{ 4 }{ x }+\frac{ 1 }{ x^{2} } ) dx$ =$\int\limits_{}^{}xdx+\int\limits_{}^{}5dx+4\int\limits_{}^{}\frac{ 1 }{ x } dx+\int\limits_{}^{}\frac{ 1 }{ x^{2} }dx$ $=\frac{ x^{2} }{ 2 }+5x+4\log \left| x \right|-\frac{ 1 }{ x }+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{Q}$$ $I=\int\limits_{}^{}(\sqrt{x}+\frac{ 1 }{ \sqrt{x} })^{2}dx$ $$\Large\color{blue}{Soln}$$\ $=\int\limits_{}^{}(x+\frac{ 1 }{ x }+2)dx$ $=\int\limits_{}^{} xdx+\int\limits_{}^{}\frac{ 1 }{ x}dx+2\int\limits_{}^{}1dx$ =$\frac{ x ^{2} }{ 2 }+\log \left| x \right|+2x+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{5}$$ $$\Large\color{red}{Q}$$ $I=\int\limits_{}^{}\sqrt{1-\cos2x}dx$ $$\Large\color{blue}{Soln}$$ $I=\sqrt{1-(1-2\sin ^{2}x)}dx$ $=\sqrt{2\sin^{2}x}dx$ $=\sqrt{2}\int\limits_{}^{}sinxdx$ $=-\sqrt{2}cosx+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{6.}$$ $$\Large\color{red}{Q}$$ $I=\int\limits_{}^{}\sqrt{1+\cos2x}dx$ $$\Large\color{blue}{Soln}$$ $=\int\limits_{}^{}\sqrt{1+2\cos ^{2}x-1}dx$ $=\int\limits_{}^{}\sqrt{2\cos ^{2}x}dx$ $=\sqrt{2}sinx+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{5~and~6}$$ $$\Large\color{red}{Q}$$ $I=\sqrt{1+\sin2x}dx$ $$\Large\color{blue}{Soln}$$ $I=\sqrt{\sin ^{2}x+\cos ^{2}x+2sinxcosx}dx$ $I=\int\limits_{}^{}\sqrt{(sinx+cosx)^{2}}dx$ $I=\int\limits_{}^{}(sinx+cosx)dx$ $I=\int\limits_{}^{}sinxdx+\int\limits_{}^{}cosxdx$ $=-cosx+sinx+C$

3 years ago
OpenStudy (zzr0ck3r):

Also, they make calculus books full of examples.

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{7.}$$ $$\Large\color{red}{Q}$$ $\int\limits_{}^{}\tan2x-\tan(x-\theta)-\tan(x+\theta)~dx$ $$\Large\color{blue}{Soln}$$ $=-1/2\log \cos2x+logcos(x-\theta)+logcos(x+\theta)+C$ $=1/2\log \sec2x-logsec(x-\theta)-logsec(x+\theta)+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{8.}$$ $$\Large\color{red}{Q}$$ $\int\limits_{}^{}cosa+\cot(x-a)sina~dx$ $$\Large\color{blue}{Soln}$$ $=cosa \int\limits1.dx+sina \int\limits \cot(x-a)dx$ $=xcosa+sina \log \left| \sin(x-a) \right|+C$ $=xcosa-sinalog \left| cosec(x-a) \right|+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{9.}$$ $$\Large\color{red}{Q}$$ $\int\limits \frac{ 1 }{ \sqrt{1+\cos2x} }dx$ $$\Large\color{blue}{Soln}$$ $=\frac{ 1 }{ \sqrt{2\cos ^{2}x} }dx$ $=\frac{ 1 }{ \sqrt{2} }\int\limits \frac{ 1 }{ \sqrt{\cos ^{2}x} }dx$ $=\frac{ 1 }{ \sqrt{2} }\int\limits secx dx$ //as 1/cosx=secx and the square and squareroot cancels. $=\frac{ 1 }{ \sqrt{2} }\log \left| secx+tanx \right|+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{10.}$$ $$\Large\color{red}{Q}$$ $\int\limits \frac{ 1 }{ \sqrt{1-cosx} }$ $$\Large\color{blue}{Soln}$$ $=\int\limits \frac{ 1 }{ \sqrt{2\sin ^{2}\frac{ x }{ 2 }} }dx$ $\frac{ 1 }{ \sqrt{2} }\int\limits cosec \frac{ x }{ 2 }dx$ // 1/sin= cosec and squareroot and square cancels. $=\frac{ 2 }{ \sqrt{2} }\log \left| cosec \frac{ x }{ 2 }-\cot \frac{ x }{ 2 } \right|+C$ // denominator of x is 2 so, apply second formula (2). $=\frac{ \sqrt{2} }{ 1 }\log \left| cosec \frac{ x }{ 2 }-\cot \frac{ x }{ 2 } \right|+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{11.}$$ $$\Large\color{red}{Q}$$ $\int\limits \tan ^{2}xdx$ $$\Large\color{blue}{Soln}$$ $=\int\limits \sec ^{2}x-1 dx$ $=tanx-x+C$

3 years ago
OpenStudy (arindameducationusc):

$$\Large\color{red}{12.}$$ $$\Large\color{red}{Q}$$ $\int\limits \cot ^{2}x dx$ $$\Large\color{blue}{Soln}$$ $=\int\limits cosec ^{2}x-1 dx$ $=-cotx-x+C$

3 years ago