Generating...                               ibee2023_n17

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

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

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

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

  3. Find $\displaystyle \int {\left(\sec x\right)^6\,\tan ^2x}{\;dx}$ by substituting $\displaystyle u=\tan x$

    $\displaystyle \int {u\,\left(u^2+1\right)^2}{\;du} = {{\tan ^6x}\over{6}}+{{\tan ^4x}\over{2}}+{{\tan ^2x}\over{2}} + C$ $\displaystyle \int {u^2\,\left(u^2+1\right)^2}{\;du} = {{\tan ^7x}\over{7}}+{{2\,\tan ^5x}\over{5}}+{{\tan ^3x}\over{3}} + C$ $\displaystyle \int {u^2}{\;du} = {{\tan ^3x}\over{3}} + C$ $\displaystyle \int {u^4}{\;du} = {{\tan ^5x}\over{5}} + C$

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

    $\displaystyle -\cot x-x + C$ $\displaystyle x-\cot x + C$ $\displaystyle -\cot x + C$ $\displaystyle \cot x + C$

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

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

  6. Find $\displaystyle \int {{{e^{\sqrt{x}}}\over{\sqrt{x}}}}{\;dx}$ by substituting $\displaystyle u=\sqrt{x}$

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

  7. Find $\displaystyle \int {{{\sin \sqrt{x}}\over{\sqrt{x}}}}{\;dx}$ by using substitution.

    $\displaystyle 2\,\sin \sqrt{x} + C$ $\displaystyle -2\,\sin \sqrt{x} + C$ $\displaystyle 2\,\cos \sqrt{x} + C$ $\displaystyle -2\,\cos \sqrt{x} + C$

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

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

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

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

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

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

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

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

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

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

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

    $\displaystyle \int {{{1}\over{\sin ^2\theta}}}{\;d\theta} = -{{\sqrt{1-x^2}}\over{x}} + C$ $\displaystyle \int {\cos ^2\theta\,\sin ^2\theta}{\;d\theta} = {{\arcsin x}... ...,\left(1-x^2\right)^{{{3}\over{2}}} }\over{4}}+{{x\,\sqrt{1-x^2}}\over{8}} + C$ $\displaystyle \int {{{\cos ^2\theta}\over{\sin ^2\theta}}}{\;d\theta} = -\arcsin x-{{\sqrt{1-x^2}}\over{x}} + C$ $\displaystyle \int {\sin ^2\theta}{\;d\theta} = {{\arcsin x}\over{2}}-{{x\,\sqrt{1-x^2}}\over{2}} + C$ $\displaystyle \int {\cos ^2\theta\,\sin \theta}{\;d\theta} = -{{\left(1-x^2\right)^{{{3}\over{2}}}}\over{3}} + C$

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

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

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

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


Department of Mathematics
Last modified: 2026-05-20