Generating...                               ibee2023_n12

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

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

  2. Find $\displaystyle \int {\sqrt{3\,x-5}}{\;dx}$ by using substitution.

    $\displaystyle 2\,\sqrt{3\,x-5} + C$ $\displaystyle -{{2\,\left(3\,x-5\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle -{{2\,\left(3\,x-5\right)^{{{3}\over{2}}}}\over{9}} + C$ $\displaystyle {{2\,\left(3\,x-5\right)^{{{3}\over{2}}}}\over{9}} + C$ $\displaystyle {{2\,\left(3\,x-5\right)^{{{3}\over{2}}}}\over{3}} + C$

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

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

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

    $\displaystyle -\tan x + C$ $\displaystyle \tan x + C$ $\displaystyle \tan x-x + C$ $\displaystyle \tan x+x + C$

  5. Find $\displaystyle \int {{{\left(x+1\right)^2}\over{\sqrt{x}}}}{\;dx}$.

    $\displaystyle {{2\,x^{{{5}\over{2}}}}\over{5}}+{{4\,x^{{{3}\over{2}}}}\over{3}}+2 \,\sqrt{x} + C$ $\displaystyle {{2\,x^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle {{2\,x^{{{7}\over{2}}}}\over{7}} + C$ $\displaystyle {{2\,\left(x+1\right)}\over{\sqrt{x}}}-{{\left(x+1\right)^2}\over{2 \,x^{{{3}\over{2}}}}} + C$

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

    $\displaystyle {{2\,\left(x+2\right)^{{{7}\over{2}}}}\over{7}}-{{8\,\left(x+2 \... ...{{{5}\over{2}}}}\over{5}}+{{8\,\left(x+2\right)^{{{3}\over{2 }}}}\over{3}} + C$ $\displaystyle {{2\,\left(x+2\right)^{{{5}\over{2}}}}\over{5}}-{{8\,\left(x+2 \right)^{{{3}\over{2}}}}\over{3}}+8\,\sqrt{x+2} + C$ $\displaystyle 2\,\sqrt{x+2} + C$ $\displaystyle {{2\,\left(x+2\right)^{{{5}\over{2}}}}\over{5}}-{{4\,\left(x+2 \right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle {{2\,\left(x+2\right)^{{{3}\over{2}}}}\over{3}}-4\,\sqrt{x+2} + C$ $\displaystyle {{2\,\left(x+2\right)^{{{3}\over{2}}}}\over{3}} + C$

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

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

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

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

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

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

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

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

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

    $\displaystyle \int {u^4}{\;du} = {{\left(x^3+2\right)^5}\over{5}} + C$ $\displaystyle \int {u^5}{\;du} = {{\left(x^3+2\right)^6}\over{6}} + C$ $\displaystyle {{\int {\left(u-2\right)\,u^5}{\;du}}\over{3}} = {{\left(x^3+2\right)^7}\over{21}}-{{\left(x^3+2\right)^6}\over{9}} + C$ $\displaystyle {{\int {u^5}{\;du}}\over{3}} = {{\left(x^3+2\right)^6}\over{18}} + C$

  12. Find $\displaystyle -\int {{{x^2}\over{x^4-16}}}{\;dx}$

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

  13. Find $\displaystyle \int {\cos ^3x}{\;dx}$ .

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

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

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

  15. Find $\displaystyle \int {{{1}\over{\sqrt{x^2-4}}}}{\;dx}$.

    $\displaystyle {\rm asinh}\; \left({{x}\over{2}}\right) + C$ $\displaystyle {{\ln \left(x+2\right)}\over{4}}-{{\ln \left(x-2\right)}\over{4}} + C$ $\displaystyle \ln \left(2\,\sqrt{x^2-4}+2\,x\right) + C$ $\displaystyle {{\ln \left(x^2+4\right)}\over{2}} + C$ $\displaystyle {{\arctan \left({{x}\over{2}}\right)}\over{2}} + C$ $\displaystyle \arcsin \left({{x}\over{2}}\right) + C$


Department of Mathematics
Last modified: 2025-10-31