Generating...                               ibee2025_n1

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

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

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

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

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

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

  4. Find $\displaystyle \int {\cos ^5x\,\sin ^2x}{\;dx}$ by using substitution.

    $\displaystyle {{\sin ^3x}\over{3}} + C$ $\displaystyle {{\sin ^6x}\over{6}}-{{\sin ^4x}\over{2}}+{{\sin ^2x}\over{2}} + C$ $\displaystyle {{\sin ^7x}\over{7}}-{{2\,\sin ^5x}\over{5}}+{{\sin ^3x}\over{3}} + C$ $\displaystyle {{\sin ^5x}\over{5}} + C$

  5. Find $\displaystyle \int {e^{x}\,\left(e^{x}+1\right)^2}{\;dx}$ by substituting $\displaystyle u=e^{x}+1$

    $\displaystyle \int {\left(u-1\right)\,u^2}{\;du} = {{\left(e^{x}+1\right)^4}\over{4}}-{{\left(e^{x}+1\right)^3}\over{3 }} + C$ $\displaystyle \int {u^2}{\;du} = {{\left(e^{x}+1\right)^3}\over{3}} + C$ $\displaystyle -\int {u^2}{\;du} = -{{\left(e^{x}+1\right)^3}\over{3}} + C$ $\displaystyle \int {u}{\;du} = {{\left(e^{x}+1\right)^2}\over{2}} + C$


Department of Mathematics
Last modified: 2026-07-16