1. Evaluate $\displaystyle \int_{0}^{\pi}{\sin x-4\,\cos x\;dx}$.

    $\displaystyle\Big[ \cos x-4\,\sin x \Big]_0^\pi = -2$ $\displaystyle\Big[ 4\,\sin x-\cos x \Big]_0^\pi = 2$ $\displaystyle\Big[ \sin x-4\,\cos x \Big]_0^\pi = 8$ $\displaystyle\Big[ -4\,\sin x-\cos x \Big]_0^\pi = 2$ $\displaystyle\Big[ 4\,\sin x+\cos x \Big]_0^\pi = -2$

  2. Find the derivative $f'(x)$ for $f(x) = \displaystyle \int_{x}^{4}{t^2\,\sin t\;dt}$.

    $f'(x) = \displaystyle x^2\,\cos x$ $f'(x) = \displaystyle -x^2\,\cos x$ $f'(x) = \displaystyle -x^2\,\sin x$ $f'(x) = \displaystyle x^2\,\sin x$

  3. Find $\displaystyle \int {\left(\sec x\right)^2\,\tan x}{\;dx}$ by using substitution.

    $\displaystyle {{1}\over{\cos x}} + C$ $\displaystyle \ln \cos x + C$ $\displaystyle {{\tan ^2x}\over{2}} + C$ $\displaystyle -{{1}\over{\cos x}} + C$

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

    $\displaystyle {{\left(x+1\right)^2}\over{2\,\sqrt{x}}}+2\,\sqrt{x}\,\left(x+1 \right) + C$ $\displaystyle {{2\,x^{{{7}\over{2}}}}\over{7}}+{{4\,x^{{{5}\over{2}}}}\over{5}}+ {{2\,x^{{{3}\over{2}}}}\over{3}} + C$ $\displaystyle {{2\,x^{{{9}\over{2}}}}\over{9}} + C$ $\displaystyle {{2\,x^{{{5}\over{2}}}}\over{5}} + C$

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

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

  6. Find $\displaystyle \int {{{1-\sqrt{x}}\over{\sqrt{x}}}}{\;dx}$.

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

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

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

  8. Evaluate $\displaystyle \int_{-2}^{-1}{e^{x+2}\;dx}$.

    $\displaystyle\Big[ e^{x+3} \Big]_{ -2}^{ -1} = e^2-e$ $\displaystyle\Big[ e^{x+2} \Big]_{ -2}^{ -1} = e-1$ $\displaystyle\Big[ 2\,e^{x} \Big]_{ -2}^{ -1} = 2\,e^ {- 1 }-2\,e^ {- 2 }$ $\displaystyle\Big[ e^{x} \Big]_{ -2}^{ -1} = e^ {- 1 }-e^ {- 2 }$

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

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

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

    $\displaystyle \int {u^5}{\;du} = {{\left(x^2+2\right)^6}\over{6}} + C$ $\displaystyle {{\int {\left(u-2\right)\,u^5}{\;du}}\over{2}} = {{\left(x^2+2\right)^7}\over{14}}-{{\left(x^2+2\right)^6}\over{6}} + C$ $\displaystyle {{\int {u^5}{\;du}}\over{2}} = {{\left(x^2+2\right)^6}\over{12}} + C$ $\displaystyle \int {u^4}{\;du} = {{\left(x^2+2\right)^5}\over{5}} + C$


Department of Mathematics
Last modified: 2026-04-14