Generating...                               ibee2023_n3

  1. Find $\displaystyle \int {x^2\,\sin x^3}{\;dx}$ by using substitution.

    $\displaystyle -{{\sin x^3}\over{3}} + C$ $\displaystyle {{\sin x^3}\over{3}} + C$ $\displaystyle {{\cos x^3}\over{3}} + C$ $\displaystyle -{{\cos x^3}\over{3}} + C$

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

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

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

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

  4. Find $\displaystyle \int {x^2\,\left(x^3+3\right)^5}{\;dx}$ by substituting $\displaystyle u=x^3+3$

    $\displaystyle \int {u^5}{\;du} = {{\left(x^3+3\right)^6}\over{6}} + C$ $\displaystyle \int {u^4}{\;du} = {{\left(x^3+3\right)^5}\over{5}} + C$ $\displaystyle {{\int {u^5}{\;du}}\over{3}} = {{\left(x^3+3\right)^6}\over{18}} + C$ $\displaystyle {{\int {\left(u-3\right)\,u^5}{\;du}}\over{3}} = {{\left(x^3+3\right)^7}\over{21}}-{{\left(x^3+3\right)^6}\over{6}} + C$

  5. Find $\displaystyle \int {\sqrt{\sec x}\,\tan x}{\;dx}$ by using substitution.

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

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

    $\displaystyle {{\int {{{\sec \theta}\over{\tan ^2\theta}}}{\;d\theta} }\over{4}} = -{{\sqrt{x^2+4}}\over{4\,x}} + C$ $\displaystyle 2\,\int {\tan ^2\theta}{\;d\theta} = x-2\,\arctan \left({{x}\over{2}}\right) + C$ $\displaystyle 2\,\int {\sec \theta\,\tan \theta}{\;d\theta} = \sqrt{x^2+4} + C$ $\displaystyle 8\,\int {\left(\sec \theta\right)^3\,\tan \theta}{\;d \theta} = {{\left(x^2+4\right)^{{{3}\over{2}}}}\over{3}} + C$

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

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

  8. Find $\displaystyle \int {\sqrt{x}\,\left(x+1\right)}{\;dx}$.

    $\displaystyle {{2\,x^{{{7}\over{2}}}}\over{7}} + C$ $\displaystyle {{2\,x^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle {{x+1}\over{2\,\sqrt{x}}}+\sqrt{x} + C$ $\displaystyle {{2\,x^{{{5}\over{2}}}}\over{5}}+{{2\,x^{{{3}\over{2}}}}\over{3}} + C$

  9. Find $\displaystyle \int {{{\left(\ln x\right)^2}\over{x}}}{\;dx}$ by substituting $\displaystyle u=\ln x$

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

  10. Find $\displaystyle \int {{{e^{\sqrt{x}}}\over{\sqrt{x}}}}{\;dx}$ by using substitution.

    $\displaystyle -x + C$ $\displaystyle x + C$ $\displaystyle -2\,e^{\sqrt{x}} + C$ $\displaystyle 2\,e^{\sqrt{x}} + C$

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

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

  12. Find $\displaystyle \int {\left(\sec x\right)^2\,\tan ^5x}{\;dx}$ by substituting $\displaystyle u=\sec x$

    $\displaystyle \int {u}{\;du} = {{\left(\sec x\right)^2}\over{2}} + C$ $\displaystyle \int {\left(u^2-1\right)^2}{\;du} = {{\left(\sec x\right)^5}\over{5}}-{{2\,\left(\sec x\right)^3}\over{ 3}}+\sec x + C$ $\displaystyle \int {u\,\left(u^2-1\right)^2}{\;du} = {{\left(\sec x\right)^... ...r{6}}-{{\left(\sec x\right)^4}\over{2}} +{{\left(\sec x\right)^2}\over{2}} + C$ $\displaystyle \int {u^3}{\;du} = {{\left(\sec x\right)^4}\over{4}} + C$

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

    $\displaystyle \left(x-1\right)\,e^{x} + C$ $\displaystyle {{e^{x}\,\left(\sin x-\cos x\right)}\over{2}} + C$ $\displaystyle \left(-x-1\right)\,e^ {- x } + C$ $\displaystyle \left(x^2-2\,x+2\right)\,e^{x} + C$ $\displaystyle \left(-x^2-2\,x-2\right)\,e^ {- x } + C$ $\displaystyle {{e^{x}\,\left(\sin x+\cos x\right)}\over{2}} + C$

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

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

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

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


Department of Mathematics
Last modified: 2026-03-24