STQM$1013$ Calculus I

Assignment $($Week $11)$: Techniques of Integration Instruction: Answer all the questions.

Evaluate the integrals using appropriate substitutions.

$($a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^2+1}{\sqrt[\phantom{2}]{x^3+3x}}dx$

$($b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{sin(\frac{1}{x})}{3x^2}dx$

$($c$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{e^x}{1+e^{2x}}dx$

$($d$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{dx}{\sqrt[\phantom{2}]{x}{e}^{(2\sqrt[\phantom{2}]{x})}}$

Determine the integral.

$($a$){\displaystyle \underset{0}{\overset{1}{}}}x{e}^{-5x}dx$

$($b$){\displaystyle \underset{\sqrt[\phantom{2}]{e}}{\overset{e}{}}}\frac{lnx}{x^2}dx$

$($c$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}{e}^{\sqrt[\phantom{2}]{x}}dx$

$($d$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}cosx\sqrt[\phantom{2}]{x}dx$

Use partial fractions to solve the indefinite integral.

$($a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{2x^3-4x^2-15x+5}{x^2-2x-8}dx$

$($b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{8x}{x^3+x^2-x-1}dx$

$($c$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}\frac{x^2}{x^4-2x^2-8}dx$

Find the indefinite integral.

$($a$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}si{n}^{3}xco{s}^{2}xdx$

$($b$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}8xco{s}^{2}xdx$

$($c$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}ta{n}^{3}2tse{c}^{3}2tdt$

$($d$){\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}se{c}^{6}4xtan4xdx$

The inner product of two functions $f$ and $g$ on $a,b$ is given by

$($:$f,g$:$)={\displaystyle \underset{a}{\overset{b}{}}}f(x)g(x)dx$

Two distinct functions $f$ and $g$ are said to be orthogonal if $($:$f,g$:$)=0.$ Show that the following set of functions,

$\{sinx,sin2x,sin3x,$dots,$cosx,cos2x,cos3x,$dots$\}$

is orthogonal on $-,.$