Generating...                               ibee2025_n15

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

    $\displaystyle -{{\sin x^3}\over{3}} + C$ $\displaystyle {{\sin x^3}\over{3}} + C$ $\displaystyle -{{\cos x^3}\over{3}} + C$ $\displaystyle {{\cos x^3}\over{3}} + C$

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

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

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

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

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

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

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

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


Department of Mathematics
Last modified: 2026-10-28