Generating...                               ibee2023_n20


You have 4 minutes. Click the button at the bottom.
  1. Find $\displaystyle \int {{{x+1}\over{\sqrt{x}}}}{\;dx}$.

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

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

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

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

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

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

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

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

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

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

    $\displaystyle
8\,\int {\left(\sec \theta\right)^2\,\tan ^2\theta}{\;d
\theta}
= {{\left(x^2-4\right)^{{{3}\over{2}}}}\over{3}} + C$
    $\displaystyle
\int {\sec \theta}{\;d\theta}
= \ln \left(2\,\sqrt{x^2-4}+2\,x\right) + C$
    $\displaystyle
2\,\int {\left(\sec \theta\right)^2}{\;d\theta}
= \sqrt{x^2-4} + C$
    $\displaystyle
{{\int {{{1}\over{\sec \theta}}}{\;d\theta}}\over{4}}
= {{\sqrt{x^2-4}}\over{4\,x}} + C$

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

    $\displaystyle
{{\cos ^3x}\over{3}}-{{\cos ^5x}\over{5}} + C$
    $\displaystyle
{{\cos ^3x}\over{3}} + C$
    $\displaystyle
{{\cos ^2x}\over{2}}-{{\cos ^4x}\over{4}} + C$
    $\displaystyle
{{\cos ^4x}\over{4}} + C$

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

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

  9. Find $\displaystyle \int {\cos ^5x\,\sin x}{\;dx}$ by substituting $\displaystyle u=\cos x$

    $\displaystyle
-\int {u^5}{\;du}
= -{{\cos ^6x}\over{6}} + C$
    $\displaystyle
\int {u^5}{\;du}
= {{\cos ^6x}\over{6}} + C$
    $\displaystyle
-\int {u^5\,\sqrt{1-u^2}}{\;du}
= {{\cos ^4x\,\left(1-\cos ^2...
...{2}}}}\over{35}}+{{8\,
\left(1-\cos ^2x\right)^{{{3}\over{2}}}}\over{105}} + C$
    $\displaystyle
-\int {u^4}{\;du}
= -{{\cos ^5x}\over{5}} + C$

  10. Find $\displaystyle \int {x^2\,\sqrt{x^2+4}}{\;dx}$ by substituting $\displaystyle x=2\,\sinh \theta$

    $\displaystyle
{{\int {{{1}\over{\sinh ^2\theta}}}{\;d\theta}}\over{4}}
= -{{\sqrt{x^2+4}}\over{4\,x}} + C$
    $\displaystyle
2\,\int {\sinh \theta}{\;d\theta}
= \sqrt{x^2+4} + C$
    $\displaystyle
\int {{{\cosh ^2\theta}\over{\sinh ^2\theta}}}{\;d\theta}
= {\rm asinh}\; \left({{x}\over{2}}\right)-{{\sqrt{x^2+4}}\over{x}} + C$
    $\displaystyle
8\,\int {\cosh ^2\theta\,\sinh \theta}{\;d\theta}
= {{\left(x^2+4\right)^{{{3}\over{2}}}}\over{3}} + C$
    $\displaystyle
16\,\int {\cosh ^2\theta\,\sinh ^2\theta}{\;d\theta}
= {{x\,\...
...}-{{x\,\sqrt{x^2+4}
}\over{2}}-2\,{\rm asinh}\; \left({{x}\over{2}}\right) + C$

  11. Find $\displaystyle \int {{{\sqrt{\ln x}}\over{x}}}{\;dx}$ by substituting $\displaystyle u=\ln x$

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

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

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

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

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

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

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

  15. Find $\displaystyle \int {\sin ^2x}{\;dx}$ .

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



Department of Mathematics
Last modified: 2025-07-22