Generating...                               ibee2023_n15

  1. Find $\displaystyle \int {{{1}\over{x^2\,\sqrt{9-x^2}}}}{\;dx}$ by substituting $\displaystyle x=3\,\sin \theta$

    $\displaystyle 81\,\int {\cos ^2\theta\,\sin ^2\theta}{\;d\theta} = -{{x\,\l... ...\sqrt{9- x^2}}\over{8}}+{{81\,\arcsin \left({{x}\over{3}}\right)}\over{8}} + C$ $\displaystyle 27\,\int {\cos ^2\theta\,\sin \theta}{\;d\theta} = -{{\left(9-x^2\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle {{\int {{{1}\over{\sin ^2\theta}}}{\;d\theta}}\over{9}} = -{{\sqrt{9-x^2}}\over{9\,x}} + C$ $\displaystyle 9\,\int {\sin ^2\theta}{\;d\theta} = {{9\,\arcsin \left({{x}\over{3}}\right)}\over{2}}-{{x\,\sqrt{9-x^2} }\over{2}} + C$ $\displaystyle 3\,\int {\sin \theta}{\;d\theta} = -\sqrt{9-x^2} + C$

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

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

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

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

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

    $\displaystyle {{\int {u^3}{\;du}}\over{4}} = {{\left(x^4+2\right)^4}\over{16}} + C$ $\displaystyle \int {u^3}{\;du} = {{\left(x^4+2\right)^4}\over{4}} + C$ $\displaystyle {{\int {\left(u-2\right)\,u^3}{\;du}}\over{4}} = {{\left(x^4+2\right)^5}\over{20}}-{{\left(x^4+2\right)^4}\over{8}} + C$ $\displaystyle \int {u^2}{\;du} = {{\left(x^4+2\right)^3}\over{3}} + C$

  5. Find $\displaystyle \int {\left(\sec x\right)^2}{\;dx}$ .

    $\displaystyle \tan x + C$ $\displaystyle \tan x-x + C$ $\displaystyle \ln \left(\tan x+\sec x\right) + C$ $\displaystyle \ln \sec x + C$

  6. Find $\displaystyle \int {\ln x}{\;dx}$ by using integration by parts.

    $\displaystyle {{x^2\,\ln x}\over{2}}-{{x^2}\over{4}} + C$ $\displaystyle -{{\ln x}\over{x}}-{{1}\over{x}} + C$ $\displaystyle x\,\ln x-x + C$ $\displaystyle {{x^3\,\ln x}\over{3}}-{{x^3}\over{9}} + C$ $\displaystyle 2\,\sqrt{x}\,\ln x-4\,\sqrt{x} + C$ $\displaystyle {{2\,x^{{{3}\over{2}}}\,\ln x}\over{3}}-{{4\,x^{{{3}\over{2}}} }\over{9}} + C$

  7. Find $\displaystyle \int {x\,\sin x}{\;dx}$.

    $\displaystyle -\ln \left(\csc x+\cot x\right) + C$ $\displaystyle {{\sin \left(2\,x\right)-2\,x\,\cos \left(2\,x\right)}\over{8}} + C$ $\displaystyle -{{\cos ^2x}\over{2}} + C$ $\displaystyle x\,\sin x+\cos x + C$ $\displaystyle \ln \left(\tan x+\sec x\right) + C$ $\displaystyle \sin x-x\,\cos x + C$

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

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

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

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

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

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

  11. Find $\displaystyle \int {\sin ^4x}{\;dx}$ .

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

  12. Find $\displaystyle \int {\sqrt{x-1}\,x^2}{\;dx}$ by substituting u = x − 1

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

  13. Find $\displaystyle \int {{{1}\over{x^2\,\sqrt{x^2-9}}}}{\;dx}$ by substituting $\displaystyle x=3\,\sec \theta$

    $\displaystyle {{\int {{{1}\over{\sec \theta}}}{\;d\theta}}\over{9}} = {{\sqrt{x^2-9}}\over{9\,x}} + C$ $\displaystyle 27\,\int {\left(\sec \theta\right)^2\,\tan ^2\theta}{\;d \theta} = {{\left(x^2-9\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle 3\,\int {\left(\sec \theta\right)^2}{\;d\theta} = \sqrt{x^2-9} + C$ $\displaystyle \int {\sec \theta}{\;d\theta} = \ln \left(2\,\sqrt{x^2-9}+2\,x\right) + C$

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

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

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

    $\displaystyle -{{1}\over{2\,x^2}} + C$ $\displaystyle -e^{{{1}\over{x}}} + C$ $\displaystyle {{1}\over{2\,x^2}} + C$ $\displaystyle e^{{{1}\over{x}}} + C$


Department of Mathematics
Last modified: 2025-09-14