Question

Solve the following

$\int \: \frac{(x + 1) {e}^{x} }{ {sin}^{2} (x {e}^{x} )} \:d x$
​

Answer 1

Important topic :-

The following question is related to topics trigonometry for this topic it is very necessary to learn Pythagoras theorem and the table value of trigonometry ratios of angles 0° , 30°, 45°, 60°, 90° {which is shared in following image}

Given :-

• $\sf \: \int \: \frac{(x + 1) {e}^{x} }{ {sin}^{2} (x {e}^{x} )} \:d x$

Solution :-

$\sf \: = let \: {xe}^{x} = t$

$\sf \: = ( {e}^{x} + {xe}^{x} )dx = dt$

$\sf \: = {e}^{x} (1 + x)dx = dt$

$\sf \: = \int \: \frac{dt}{ {sin}^{2} t} \: = \int \: cose \: {c}^{2} \: tdt$

$\sf \: = - cot(t) + c$

$\sf \: = - cot(x {e}^{x} ) + c$

$\sf \: = \int \: \frac{(x + 1) {e}^{x} }{ {sin}^{2} (x {e}^{x} )} \:d x = - \cot(x {e}^{x} ) + c$

$\sf \fbox \red{Answer } = \: \: \sf \int \: \frac{(x + 1) {e}^{x} }{ {sin}^{2} (x {e}^{x} )} \:d x = \sf \red{ - \cot(x {e}^{x} ) + c}$

Note :-

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Addictonal information :-

There are 6 trigonometric functions which are:

• Sine function
• Cosine function
• Tan function
• Sec function
• Cot function
• Cosec function

warning :-

If you are writing this in you class book please don't write Important topic, Note and addictonal information.

Answer 2

$\large\underline{\sf{Solution-}}$

Given integral is

$\rm \: \displaystyle\int\rm \frac{(x + 1){e}^{x}}{sin^{2} (x{e}^{x})} \: dx$

To evaluate this integral, we use method of Substitution.

So, Substitute

$\red{\rm \: x{e}^{x} = y \: }$

On differentiating both sides w. r. t. x, we get

$\red{\rm \: x{e}^{x} + {e}^{x} \times 1 = \dfrac{dy}{dx} \: }$

$\red{\rm \: x{e}^{x} + {e}^{x} = \dfrac{dy}{dx} \: }$

$\red{\rm \: (x +1) {e}^{x} = \dfrac{dy}{dx} \: }$

$\red{\rm \: (x +1) {e}^{x}dx = dy \: }$

So, on substituting these values in above integral, we get

$\rm \: = \: \displaystyle\int\rm \frac{dy}{ {sin}^{2} y}$

$\rm \: = \: \displaystyle\int\rm {cosec}^{2}y \: dy$

$\rm \: = \: - \: coty \: + \: c$

$\rm \: = \: - \: cot(x{e}^{x}) \: + \: c$

Hence,

$\rm\implies \:\boxed{ \rm{ \:\displaystyle\int\rm \frac{(x + 1){e}^{x}}{ {sin}^{2}(x{e}^{x}) } dx = \: - \: cot(x{e}^{x}) \: + \: c \: }}$

$\rule{190pt}{2pt}$

Additional Information :-

$\begin{gathered}\begin{gathered}\boxed{\begin{array}{c|c} \bf f(x) & \bf \displaystyle \int \rm \:f(x) \: dx\\ \\ \frac{\qquad \qquad}{} & \frac{\qquad \qquad}{} \\ \sf k & \sf kx + c \\ \\ \sf sinx & \sf - \: cosx+ c \\ \\ \sf cosx & \sf \: sinx + c\\ \\ \sf {sec}^{2} x & \sf tanx + c\\ \\ \sf {cosec}^{2}x & \sf - cotx+ c \\ \\ \sf secx \: tanx & \sf secx + c\\ \\ \sf cosecx \: cotx& \sf - \: cosecx + c\\ \\ \sf tanx & \sf logsecx + c\\ \\ \sf \dfrac{1}{x} & \sf logx+ c\\ \\ \sf {e}^{x} & \sf {e}^{x} + c\end{array}} \\ \end{gathered}\end{gathered}$