Generating...                               ibee2023_n16

  1. Find $\displaystyle -\int {{{\sqrt{1-x^2}\,\left(\arcsin x+1\right)}\over{x^2-1}}}{\;dx }$

    $\displaystyle 2\,\sqrt{\arcsin x+1} + C$ $\displaystyle \arcsin x + C$ $\displaystyle {{2\,\left(\arcsin x+1\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle \ln \left(\arcsin x+1\right) + C$ $\displaystyle {{\left(\arcsin x+1\right)^2}\over{2}} + C$

  2. Find $\displaystyle \int {x^2\,e^{x}}{\;dx}$ by using integration by parts.

    $\displaystyle \left(-x^2-2\,x-2\right)\,e^ {- x } + C$ $\displaystyle -{{2\,e^{x}\,\sin \left(2\,x\right)+e^{x}\,\cos \left(2\,x\right)-5 \,e^{x}}\over{10}} + C$ $\displaystyle {{2\,e^{x}\,\sin \left(2\,x\right)+e^{x}\,\cos \left(2\,x\right)+5 \,e^{x}}\over{10}} + C$ $\displaystyle \left(x^2-2\,x+2\right)\,e^{x} + C$ $\displaystyle {{e^{x}\,\left(\sin \left(2\,x\right)-2\,\cos \left(2\,x\right) \right)}\over{10}} + C$

  3. Find $\displaystyle \int {\left(\sec x\right)^2\,\tan x}{\;dx}$ by using substitution.

    $\displaystyle {{\tan ^2x}\over{2}} + C$ $\displaystyle -\sec x + C$ $\displaystyle \sec x + C$ $\displaystyle \ln \cos x + C$

  4. Find $\displaystyle -\int {{{x^2}\over{x^4-16}}}{\;dx}$

    $\displaystyle {{{\rm atanh}\; \left({{x}\over{2}}\right)}\over{4}}-{{\arctan \left({{x}\over{2}}\right)}\over{4}} + C$ $\displaystyle {{{\rm atanh}\; \left({{x}\over{2}}\right)}\over{16}}+{{\arctan \left({{x}\over{2}}\right)}\over{16}} + C$ $\displaystyle {{{\rm atanh}\; x}\over{2}}-{{\arctan x}\over{2}} + C$ $\displaystyle {{{\rm atanh}\; x}\over{2}}+{{\arctan x}\over{2}} + C$

  5. Find $\displaystyle \int {{{x^2}\over{\sqrt{x+3}}}}{\;dx}$.

    $\displaystyle {{2\,\left(x+3\right)^{{{3}\over{2}}}}\over{3}}-6\,\sqrt{x+3} + C$ $\displaystyle {{2\,\left(x+3\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle 2\,\sqrt{x+3} + C$ $\displaystyle {{2\,\left(x+3\right)^{{{5}\over{2}}}}\over{5}}-2\,\left(x+3\right) ^{{{3}\over{2}}} + C$ $\displaystyle {{2\,\left(x+3\right)^{{{7}\over{2}}}}\over{7}}-{{12\,\left(x+3 \right)^{{{5}\over{2}}}}\over{5}}+6\,\left(x+3\right)^{{{3}\over{2}} } + C$ $\displaystyle {{2\,\left(x+3\right)^{{{5}\over{2}}}}\over{5}}-4\,\left(x+3\right) ^{{{3}\over{2}}}+18\,\sqrt{x+3} + C$

  6. Find $\displaystyle \int {{{x+1}\over{x^2+1}}}{\;dx}$ .

    $\displaystyle {{\ln \left(x^2+1\right)}\over{2}}+2\,\arctan x + C$ $\displaystyle \ln \left(x^2+1\right)+\arctan x + C$ $\displaystyle {{\ln \left(x^2+1\right)}\over{2}} + C$ $\displaystyle {{\ln \left(x^2+1\right)}\over{2}}+\arctan x + C$ $\displaystyle \arctan x-{{\ln \left(x^2+1\right)}\over{2}} + C$

  7. Find $\displaystyle \int {{{x^2+2}\over{x^2+1}}}{\;dx}$.

    $\displaystyle \arctan x+x + C$ $\displaystyle -{{\ln \left(x+1\right)}\over{2}}+x+{{\ln \left(x-1\right)}\over{ 2}} + C$ $\displaystyle x-{{1}\over{x}} + C$ $\displaystyle -{{2\,x}\over{\left(x^2+1\right)^2}} + C$

  8. Find $\displaystyle \int {{{2\,x^2-6\,x+5}\over{x-2}}}{\;dx}$ .

    $\displaystyle {{x^2}\over{2}}+2\,x+2\,\ln \left(x-2\right) + C$ $\displaystyle {{x^2}\over{2}}-x+2\,\ln \left(x-2\right) + C$ $\displaystyle x^2+2\,x+\ln \left(x-2\right) + C$ $\displaystyle x^2+4\,x+\ln \left(x-2\right) + C$ $\displaystyle x^2-2\,x+\ln \left(x-2\right) + C$ $\displaystyle {{x^2}\over{2}}+x+2\,\ln \left(x-2\right) + C$

  9. Find $\displaystyle \int {\cos ^4x}{\;dx}$ .

    $\displaystyle {{\sin \left(4\,x\right)}\over{32}}-{{\sin \left(2\,x\right)}\over{ 4}}+{{3\,x}\over{8}} + C$ $\displaystyle {{\sin \left(4\,x\right)}\over{32}}+{{\sin \left(2\,x\right)}\over{ 4}}+{{3\,x}\over{8}} + C$ $\displaystyle \sin x-{{\sin ^3x}\over{3}} + C$ $\displaystyle {{\sin \left(2\,x\right)}\over{4}}+{{x}\over{2}} + C$ $\displaystyle {{\cos ^3x}\over{3}}-\cos x + C$ $\displaystyle {{x}\over{2}}-{{\sin \left(2\,x\right)}\over{4}} + C$

  10. Find $\displaystyle \int {{{x^2+x+1}\over{x^3+x}}}{\;dx}$ .

    $\displaystyle {{\ln \left(x^2+1\right)}\over{2}}+\ln x-\arctan x + C$ $\displaystyle {{\ln \left(x^2+1\right)}\over{2}}+\ln x+\arctan x + C$ $\displaystyle \ln x+\arctan x + C$ $\displaystyle \ln x-\arctan x + C$ $\displaystyle {{\ln \left(x^2+1\right)}\over{2}}-\arctan x + C$

  11. Find $\displaystyle \int {{{x^2}\over{\sqrt{x^2+4}}}}{\;dx}$ by substituting $\displaystyle x=2\,\sinh \theta$

    $\displaystyle 8\,\int {\cosh ^2\theta\,\sinh \theta}{\;d\theta} = {{\left(x^2+4\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle \int {{{\cosh ^2\theta}\over{\sinh ^2\theta}}}{\;d\theta} = {\rm asinh}\; \left({{x}\over{2}}\right)-{{\sqrt{x^2+4}}\over{x}} + C$ $\displaystyle {{\int {{{1}\over{\sinh ^2\theta}}}{\;d\theta}}\over{4}} = -{{\sqrt{x^2+4}}\over{4\,x}} + C$ $\displaystyle 2\,\int {\sinh \theta}{\;d\theta} = \sqrt{x^2+4} + C$ $\displaystyle 4\,\int {\sinh ^2\theta}{\;d\theta} = {{x\,\sqrt{x^2+4}}\over{2}}-2\,{\rm asinh}\; \left({{x}\over{2}} \right) + C$

  12. Find $\displaystyle \int {{{1}\over{x^2+9}}}{\;dx}$.

    $\displaystyle {{\ln \left(x+3\right)}\over{6}}-{{\ln \left(x-3\right)}\over{6}} + C$ $\displaystyle \ln \left(2\,\sqrt{x^2-9}+2\,x\right) + C$ $\displaystyle {{\arctan \left({{x}\over{3}}\right)}\over{3}} + C$ $\displaystyle {{\ln \left(x^2+9\right)}\over{2}} + C$ $\displaystyle \arcsin \left({{x}\over{3}}\right) + C$ $\displaystyle {\rm asinh}\; \left({{x}\over{3}}\right) + C$

  13. Find $\displaystyle \int {x\,\sqrt{x^2+1}}{\;dx}$ by substituting $\displaystyle x=\tan \theta$

    $\displaystyle \int {\sec \theta\,\tan \theta}{\;d\theta} = \sqrt{x^2+1} + C$ $\displaystyle \int {\tan ^2\theta}{\;d\theta} = x-\arctan x + C$ $\displaystyle \int {\left(\sec \theta\right)^3\,\tan \theta}{\;d\theta} = {{\left(x^2+1\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle \int {{{\sec \theta}\over{\tan ^2\theta}}}{\;d\theta} = -{{\sqrt{x^2+1}}\over{x}} + C$

  14. Find $\displaystyle \int {\cos ^5x\,\sin x}{\;dx}$ by using substitution.

    $\displaystyle {{\sin ^2x}\over{2}} + C$ $\displaystyle {{\sin ^4x}\over{4}} + C$ $\displaystyle -{{\cos ^6x}\over{6}} + C$ $\displaystyle {{\sin ^5x}\over{5}}-{{2\,\sin ^3x}\over{3}}+\sin x + C$

  15. Find $\displaystyle \int {{{3\,x-5}\over{x^2-3\,x+2}}}{\;dx}$ .

    $\displaystyle \ln \left(x+1\right)+2\,\ln \left(x-1\right) + C$ $\displaystyle 2\,\ln \left(x+2\right)+\ln \left(x-1\right) + C$ $\displaystyle \ln \left(x-1\right)+2\,\ln \left(x-2\right) + C$ $\displaystyle 2\,\ln \left(x+1\right)+\ln \left(x-1\right) + C$ $\displaystyle \ln \left(x+2\right)+2\,\ln \left(x-1\right) + C$ $\displaystyle 2\,\ln \left(x-1\right)+\ln \left(x-2\right) + C$


Department of Mathematics
Last modified: 2025-05-04