Generating...                               ibee2025_n21

  1. Find $\displaystyle \int {e^{\sqrt{x}}}{\;dx}$ by substituting $\displaystyle u=\sqrt{x}$

    $\displaystyle 2\,\int {\ln \left(u+1\right)}{\;du} = 2\,\left(\ln \left(\sqrt{x}+1\right)\,\left(\sqrt{x}+1\right)- \sqrt{x}-1\right) + C$ $\displaystyle 2\,\int {u\,\ln \left(u+1\right)}{\;du} = \ln \left(\sqrt{x}+1\right)\,x-{{x}\over{2}}+\sqrt{x}-\ln \left( \sqrt{x}+1\right) + C$ $\displaystyle 2\,\int {e^{u}}{\;du} = 2\,e^{\sqrt{x}} + C$ $\displaystyle 2\,\int {u\,e^{u}}{\;du} = 2\,\left(\sqrt{x}-1\right)\,e^{\sqrt{x}} + C$

  2. Find $\displaystyle \int {{{x}\over{\sqrt{x+1}}}}{\;dx}$ by substituting u = x + 1

    $\displaystyle \int {{{1}\over{\sqrt{u}}}}{\;du} = 2\,\sqrt{x-1} + C$ $\displaystyle \int {{{1}\over{\sqrt{u}}}}{\;du} = 2\,\sqrt{x+1} + C$ $\displaystyle \int {{{u-1}\over{\sqrt{u}}}}{\;du} = {{2\,\left(x+1\right)^{{{3}\over{2}}}}\over{3}}-2\,\sqrt{x+1} + C$ $\displaystyle \int {{{u+1}\over{\sqrt{u}}}}{\;du} = {{2\,\left(x-1\right)^{{{3}\over{2}}}}\over{3}}+2\,\sqrt{x-1} + C$

  3. Find $\displaystyle -\int {{{1}\over{x^4-1}}}{\;dx}$

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

  4. Find $\displaystyle \int {{{3\,x-2}\over{x^2-4}}}{\;dx}$ .

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

  5. Find $\displaystyle \int {{{x^2+x+9}\over{x^3+9\,x}}}{\;dx}$ .

    $\displaystyle {{\ln \left(x^2+9\right)}\over{2}}+\ln x+{{\arctan \left({{x }\over{3}}\right)}\over{3}} + C$ $\displaystyle \ln x+{{\arctan \left({{x}\over{3}}\right)}\over{3}} + C$ $\displaystyle \ln x-{{\arctan \left({{x}\over{3}}\right)}\over{3}} + C$ $\displaystyle {{\ln \left(x^2+9\right)}\over{2}}+\ln x-{{\arctan \left({{x }\over{3}}\right)}\over{3}} + C$ $\displaystyle {{\ln \left(x^2+9\right)}\over{2}}-{{\arctan \left({{x}\over{3}} \right)}\over{3}} + C$


Department of Mathematics
Last modified: 2025-06-19