Quiz

Generating...

ibee2023_n28

Find $\displaystyle \int {\left(\sec x\right)^4\,\tan x}{\;dx}$ by substituting $\displaystyle u=\tan x$
$\displaystyle \int {u^2+1}{\;du} = {{\tan ^3x}\over{3}}+\tan x + C$ $\displaystyle \int {u^2}{\;du} = {{\tan ^3x}\over{3}} + C$ $\displaystyle \int {u}{\;du} = {{\tan ^2x}\over{2}} + C$ $\displaystyle \int {u\,\left(u^2+1\right)}{\;du} = {{\tan ^4x}\over{4}}+{{\tan ^2x}\over{2}} + C$
Find $\displaystyle \int {{{x}\over{\sqrt{x^2-9}}}}{\;dx}$ by substituting $\displaystyle x=3\,\sec \theta$
$\displaystyle 27\,\int {\left(\sec \theta\right)^2\,\tan ^2\theta}{\;d \theta} = {{\left(x^2-9\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle \int {\sec \theta}{\;d\theta} = \ln \left(2\,\sqrt{x^2-9}+2\,x\right) + C$ $\displaystyle 3\,\int {\left(\sec \theta\right)^2}{\;d\theta} = \sqrt{x^2-9} + C$ $\displaystyle {{\int {{{1}\over{\sec \theta}}}{\;d\theta}}\over{9}} = {{\sqrt{x^2-9}}\over{9\,x}} + C$
Find $\displaystyle \int {{{e^{x}}\over{e^{x}+1}}}{\;dx}$ by substituting $\displaystyle u=e^{x}+1$
$\displaystyle -\int {{{1}\over{u}}}{\;du} = -\ln \left(e^{x}+1\right) + C$ $\displaystyle \int {{{1}\over{u}}}{\;du} = \ln \left(e^{x}+1\right) + C$ $\displaystyle \int {{{u-1}\over{u}}}{\;du} = -\ln \left(e^{x}+1\right)+e^{x}+1 + C$ $\displaystyle \int {{{1}\over{u^2}}}{\;du} = -{{1}\over{e^{x}+1}} + C$
Find $\displaystyle \int {\sqrt{5\,x+2}}{\;dx}$ by using substitution.
$\displaystyle {{2\,\left(5\,x+2\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle 2\,\sqrt{5\,x+2} + C$ $\displaystyle -{{2\,\left(5\,x+2\right)^{{{3}\over{2}}}}\over{15}} + C$ $\displaystyle -{{2\,\left(5\,x+2\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle {{2\,\left(5\,x+2\right)^{{{3}\over{2}}}}\over{15}} + C$
Find $\displaystyle \int {{{x+1}\over{\sqrt{x}}}}{\;dx}$ .
$\displaystyle {{2\,x^{{{3}\over{2}}}}\over{3}}+2\,\sqrt{x} + C$ $\displaystyle 2\,\sqrt{x} + C$ $\displaystyle {{2\,x^{{{5}\over{2}}}}\over{5}} + C$ $\displaystyle {{1}\over{\sqrt{x}}}-{{x+1}\over{2\,x^{{{3}\over{2}}}}} + C$
Find $\displaystyle \int {\csc x}{\;dx}$ .
$\displaystyle {{\sin \left(2\,x\right)-2\,x\,\cos \left(2\,x\right)}\over{8}} + C$ $\displaystyle \sin x-x\,\cos x + C$ $\displaystyle x\,\sin x+\cos x + C$ $\displaystyle -\ln \left(\csc x+\cot x\right) + C$ $\displaystyle \ln \left(\tan x+\sec x\right) + C$ $\displaystyle -{{\cos ^2x}\over{2}} + C$
Find $\displaystyle -\int {{{1}\over{x^4-1}}}{\;dx}$
$\displaystyle {{{\rm atanh}\; x}\over{2}}-{{\arctan x}\over{2}} + C$ $\displaystyle {{{\rm atanh}\; x}\over{2}}+{{\arctan x}\over{2}} + C$ $\displaystyle {{{\rm atanh}\; \left({{x}\over{2}}\right)}\over{16}}+{{\arctan \left({{x}\over{2}}\right)}\over{16}} + C$ $\displaystyle {{{\rm atanh}\; \left({{x}\over{2}}\right)}\over{4}}-{{\arctan \left({{x}\over{2}}\right)}\over{4}} + C$
Find $\displaystyle \int {e^{x}\,\cos ^2x}{\;dx}$ by using integration by parts.
$\displaystyle \left(-x^2-2\,x-2\right)\,e^ {- x } + C$ $\displaystyle -{{2\,e^{x}\,\sin \left(2\,x\right)+e^{x}\,\cos \left(2\,x\right)-5 \,e^{x}}\over{10}} + C$ $\displaystyle {{e^{x}\,\left(\sin \left(2\,x\right)-2\,\cos \left(2\,x\right) \right)}\over{10}} + C$ $\displaystyle \left(x^2-2\,x+2\right)\,e^{x} + C$ $\displaystyle {{2\,e^{x}\,\sin \left(2\,x\right)+e^{x}\,\cos \left(2\,x\right)+5 \,e^{x}}\over{10}} + C$
Find $\displaystyle \int {{{2\,x^2-x+4}\over{x^3+4\,x}}}{\;dx}$ .
$\displaystyle \ln x+{{\arctan \left({{x}\over{2}}\right)}\over{2}} + C$ $\displaystyle {{\ln \left(x^2+4\right)}\over{2}}+\ln x-{{\arctan \left({{x }\over{2}}\right)}\over{2}} + C$ $\displaystyle {{\ln \left(x^2+4\right)}\over{2}}+\ln x+{{\arctan \left({{x }\over{2}}\right)}\over{2}} + C$ $\displaystyle {{\arctan \left({{x}\over{2}}\right)}\over{2}} + C$ $\displaystyle \ln x-{{\arctan \left({{x}\over{2}}\right)}\over{2}} + C$
Find $\displaystyle \int {\sqrt{x+3}}{\;dx}$ .
$\displaystyle {{2\,\left(x+3\right)^{{{3}\over{2}}}}\over{3}}-6\,\sqrt{x+3} + C$ $\displaystyle {{2\,\left(x+3\right)^{{{5}\over{2}}}}\over{5}}-4\,\left(x+3\right) ^{{{3}\over{2}}}+18\,\sqrt{x+3} + C$ $\displaystyle {{2\,\left(x+3\right)^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle {{2\,\left(x+3\right)^{{{7}\over{2}}}}\over{7}}-{{12\,\left(x+3 \right)^{{{5}\over{2}}}}\over{5}}+6\,\left(x+3\right)^{{{3}\over{2}} } + C$ $\displaystyle 2\,\sqrt{x+3} + C$ $\displaystyle {{2\,\left(x+3\right)^{{{5}\over{2}}}}\over{5}}-2\,\left(x+3\right) ^{{{3}\over{2}}} + C$
Find $\displaystyle \int {\cos ^5x\,\sin ^2x}{\;dx}$ by using substitution.
$\displaystyle {{\sin ^3x}\over{3}} + C$ $\displaystyle {{\sin ^6x}\over{6}}-{{\sin ^4x}\over{2}}+{{\sin ^2x}\over{2}} + C$ $\displaystyle {{\sin ^7x}\over{7}}-{{2\,\sin ^5x}\over{5}}+{{\sin ^3x}\over{3}} + C$ $\displaystyle {{\sin ^5x}\over{5}} + C$
Find $\displaystyle \int {\tan x}{\;dx}$ .
$\displaystyle \tan x-x + C$ $\displaystyle \ln \sec x + C$ $\displaystyle \tan x + C$ $\displaystyle \ln \left(\tan x+\sec x\right) + C$
Find $\displaystyle \int {\sqrt{\sec x}\,\tan x}{\;dx}$ by using substitution.
$\displaystyle -{{2}\over{\sqrt{\sec x}}} + C$ $\displaystyle {{2}\over{3\,\left(\sec x\right)^{{{3}\over{2}}}}} + C$ $\displaystyle {{2}\over{\sqrt{\sec x}}} + C$ $\displaystyle 2\,\sqrt{\sec x} + C$
Find $\displaystyle \int {{{3\,x-4}\over{x^2-3\,x+2}}}{\;dx}$ .
$\displaystyle 2\,\ln \left(x+2\right)+\ln \left(x-2\right) + C$ $\displaystyle \ln \left(x+1\right)+2\,\ln \left(x-2\right) + C$ $\displaystyle \ln \left(x-1\right)+2\,\ln \left(x-2\right) + C$ $\displaystyle 2\,\ln \left(x-1\right)+\ln \left(x-2\right) + C$ $\displaystyle \ln \left(x+2\right)+2\,\ln \left(x-2\right) + C$ $\displaystyle 2\,\ln \left(x+1\right)+\ln \left(x-2\right) + C$
Find $\displaystyle \int {{{x^2-3\,x+4}\over{x-2}}}{\;dx}$ .
$\displaystyle {{x^2}\over{2}}+2\,x+2\,\ln \left(x-2\right) + C$ $\displaystyle x^2+4\,x+\ln \left(x-2\right) + C$ $\displaystyle {{x^2}\over{2}}+x+2\,\ln \left(x-2\right) + C$ $\displaystyle x^2+2\,x+\ln \left(x-2\right) + C$ $\displaystyle x^2-2\,x+\ln \left(x-2\right) + C$ $\displaystyle {{x^2}\over{2}}-x+2\,\ln \left(x-2\right) + C$

Department of Mathematics
Last modified: 2026-07-16