Generating...                               ibee2025_n21

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

    $\displaystyle {{x^4}\over{4}} + C$ $\displaystyle -{{e^{x^2}}\over{2}} + C$ $\displaystyle {{e^{x^2}}\over{2}} + C$ $\displaystyle -{{x^4}\over{4}} + C$

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

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

  3. Find $\displaystyle \int {\ln \left(\sqrt{x}+1\right)}{\;dx}$ by substituting $\displaystyle u=\sqrt{x}$

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

  4. Find $\displaystyle \int {{{\sqrt{x}+1}\over{\sqrt{x}}}}{\;dx}$.

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

  5. Find $\displaystyle \int {\left(2\,x-5\right)^{{{3}\over{2}}}}{\;dx}$ by using substitution.

    $\displaystyle -{{\left(2\,x-5\right)^{{{5}\over{2}}}}\over{5}} + C$ $\displaystyle {{2\,\left(2\,x-5\right)^{{{5}\over{2}}}}\over{5}} + C$ $\displaystyle -{{2\,\left(2\,x-5\right)^{{{5}\over{2}}}}\over{5}} + C$ $\displaystyle {{\left(2\,x-5\right)^{{{5}\over{2}}}}\over{5}} + C$ $\displaystyle {{2\,\left(2\,x-5\right)^{{{3}\over{2}}}}\over{3}} + C$


Department of Mathematics
Last modified: 2026-03-24