10 Integral Problems

I wrote some problems inspired by the MIT integration bee. Enjoy ^-^

  1. dxx+ln(xx) \displaystyle\int{\dfrac{dx}{x + \ln{(x^x)}}}

  2. 2x22x222xdx \displaystyle\int{2^{x} \cdot 2^{2^{x}} \cdot 2^{2^{2^{x}}}dx}

  3. 1+1+xdx \displaystyle\int{\sqrt{1+\sqrt{1+x}} dx }

  4. sin(8x)sin(x)dx \displaystyle\int{ \dfrac{\sin{(8x)}}{\sin{(x)}}dx }

  5. 1xxdx \displaystyle\int{ \dfrac{\sqrt{1-x}}{x}dx }

  6. cos1(x)2dx \displaystyle\int{ \cos^{-1}{(x)}^2dx}

  7. 122dxx4(1+x3) \displaystyle\int_{\frac{1}{2}}^{2} { \dfrac{dx}{x^4(1+x^3)} }

  8. 0πsin(x)cos(x)dx \displaystyle\int_{0}^{\pi} { \lfloor \sin{(x)} - \cos{(x)} \rfloor dx }

  9. π3π31sin(x)31sin(x)2dx \displaystyle\int_{-\frac{\pi}{3}}^{\frac{\pi}{3}} { \dfrac{1-\sqrt[3]{\sin{(x)}}}{1-\sin{(x)}^2} dx}

  10. 01x5ln(x)10dx \displaystyle\int_{0}^{1} { x^5\ln{(x)}^{10} dx}

Note by Daniel Hinds
1 month, 1 week ago

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1.) 1x(1+lnx)dx\int_{ }^{ }\frac{1}{x\left(1+\ln x\right)}dx

=ln(1+lnx)+C                    (Took 1+lnx = t )=\ln\left(\left|1+\ln x\right|\right)+C\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(Took\ 1+\ln x\ =\ t\ \right)

Aaghaz Mahajan - 1 month, 1 week ago

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2.) ln3(2)(2x22x222x)ln3(2)dx\ln^3\left(2\right)\int_{ }^{ }\frac{\left(2^x\cdot2^{2^x}\cdot2^{2^{2^x}}\right)}{\ln^3\left(2\right)}dx

=(2(22x+11))ln3(2)+C                      (Took 222x=t)=\frac{\left(2^{\left(2^{2^x+1}-1\right)}\right)}{\ln^3\left(2\right)}+C\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(Took\ 2^{2^{2^x}}=t\right)

Aaghaz Mahajan - 1 month, 1 week ago

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3.) 1+1+xdx\int_{ }^{ }\sqrt{1+\sqrt{1+x}}dx

=2(t1)tdt                      (Took 1+x=t)=2\int_{ }^{ }\left(t-1\right)\sqrt{t}dt\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(Took\ \sqrt{1+x}=t\right)

=2(2t5252t323)+C=2\left(\frac{2t^{\frac{5}{2}}}{5}-\frac{2t^{\frac{3}{2}}}{3}\right)+C

=4(1+x)5454(1+x)343+C=\frac{4\left(1+x\right)^{\frac{5}{4}}}{5}-\frac{4\left(1+x\right)^{\frac{3}{4}}}{3}+C

Aaghaz Mahajan - 1 month, 1 week ago

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5.) 2cos2tsintdt                 (Took  x = sin2t)2\int_{ }^{ }\frac{\cos^2t}{\sin t}dt\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \left(Took\ \ x\ =\ \sin^2t\right)

=2((csctdt)(sintdt))=2\left(\left(\int_{ }^{ }\csc tdt\right)-\left(\int_{ }^{ }\sin tdt\right)\right)

=2ln(csctcott)+cost+C=2\ln\left(\csc t-\cot t\right)+\cos t+C

=2ln(11x)lnx+1x+C=2\ln\left(1-\sqrt{1-x}\right)-\ln x+\sqrt{1-x}+C

Aaghaz Mahajan - 1 month, 1 week ago

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For problem 7, make the substitution u=1/x3,u = 1/x^3, du/3=dx/x4,-du/3 = dx/x^4, to get 81/8du/31+1/u=131/88uduu+1=13(818)131/88duu+1=21813(ln(9)ln(9/8))=21813ln(8)=218ln(2). \begin{aligned} \int_{8}^{1/8} \frac{-du/3}{1+1/u} = \frac13 \int_{1/8}^8 \frac{u \, du}{u+1} &= \frac13\left( 8-\frac18 \right) - \frac13 \int_{1/8}^8 \frac{du}{u+1} \\ &= \frac{21}8 - \frac13 (\ln(9) - \ln(9/8)) \\ &= \frac{21}8 - \frac13 \ln(8) \\ &= \frac{21}8 - \ln(2). \end{aligned}

Patrick Corn - 3 weeks, 1 day ago

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For problem 8, this should just be π4(1)+π4(0)+π2(1)=π4.\frac{\pi}4(-1) + \frac{\pi}4(0) + \frac{\pi}2(1) = \frac{\pi}4.

Patrick Corn - 3 weeks, 1 day ago

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Solution to problem 4: WKT sin(2kx)/sin(x) = 2[cos(x) + cos(3x) +...+ cos((2k-1)x)]

Plugging k=4 gives us the integrand.

Using the R.H.S expression, the integral is 2[ sin(x) + ((sin(3x))/3) + ((sin(5x))/5) + ((sin(7x))/7) ]1😁

Really sorry for the format.

tc adityaa - 1 month ago

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We can use

sin(2kx)sinx=2n=1kcos((2n1)x)\frac{\sin\left(2kx\right)}{\sin x}=2\sum_{n=1}^k\cos\left(\left(2n-1\right)x\right)

And thus, plugging k=4k=4 gives us the integral to be equal to

2(sinx+sin(3x)3+sin(5x)5+sin(7x)7)+C2\left(\sin x+\frac{\sin\left(3x\right)}{3}+\frac{\sin\left(5x\right)}{5}+\frac{\sin\left(7x\right)}{7}\right)+C

@tc adityaa Here, I have written it in LaTeX\LaTeX

Aaghaz Mahajan - 1 month ago

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Excuse me, I am a student currently studying in Grade 10 and I want to master calculus. I have joined some related courses on Brilliant and I have been practicing a lot. But I am still having some difficulties in distilling the fundamental concepts in calculus. What concepts in calculus do you think are essential and fundamental? What should I pay attention on in each concept? I can only think of "limit", "continuity", "derivatives", "differentiation & integration" and a few theorems(e.g. EVT, IVT). Any comment would be a great help for me.

Yung Kit So - 3 weeks, 2 days ago

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Start studying this book...

Aaghaz Mahajan - 3 weeks, 2 days ago

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Thank you!

Yung Kit So - 3 weeks, 1 day ago

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