Question

$\large\underline \red{{\bf{Question}}}$

$\large{\sf{ \int \frac{ \{x(\pi + 49) \} ^{ \frac{15}{7} } }{ {\pi}^{2}( {x}^{\pi} + 7) }dx }}$
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Answer 1

Answer:

answer is

654

⚡

that question is very difficult but important as it is

Answer 2

$\large{\underline{\underline \red{{\bf{Question : }}}}}$

$\begin{gathered}\large{\sf{ \int \frac{ \{x(\pi + 49) \} ^{ \frac{15}{7} } }{ {\pi}^{2}( {x}^{\pi} + 7) } \: dx }} \end{gathered}$

$\rm \large \underline{Solution : - }$

$\rm \bf{ we \: know \: that \: { \: \large\pi = \frac{22}{7} }}$

The π value in fraction is 22/7. It is known that pi is an irrational number which means that the digits after the decimal point are never-ending and being a non-terminating value. Therefore, 22/7 is used for everyday calculations. ‘π’ is not equal to the ratio of any two number, which makes it an irrational number.

$\begin{gathered}\sf \: {So \: let , \: t \: = \: {x \: }^{\pi} \: + \: 7 } \end{gathered}$

and differentiate with respect to get x

Now we get,

$\begin{gathered} \displaystyle \bf{t \: = {x}^{ \pi} \: + \: 7\: }\end{gathered}$

$\begin{gathered} \sf : \implies{ \: \dfrac{dt}{dx} \: = \displaystyle \pi \: x \: ^{\pi \: - \: 1} \: }\end{gathered}$

$\begin{gathered} \sf : \implies{ \: dt\: = \displaystyle \pi \: x \: ^{\pi \: - \: 1} \: dx }\end{gathered}$

$\begin{gathered} \sf : \implies{ \: dt\: = \displaystyle \pi \: x \: ^{ \frac{22}{7} \: - \: 1} \: dx}\end{gathered}$

$\begin{gathered} \sf : \implies{ \: dt\: = \displaystyle \pi \: x ^{\frac{22 - 7}{7} }\: dx \: }\end{gathered}$

$\begin{gathered} \sf : \implies{ \: dt \: = \displaystyle \pi \: x \: ^{\frac {15}{7} } \: dx}\end{gathered}$

$\begin{gathered} \sf : \implies{ \: dx\: = \frac{dt}{ \displaystyle{\pi \: x}^{ \frac{15}{7} } } }\end{gathered}$

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Given function —

$\begin{gathered} \qquad \bigstar \: \: \boxed{\large{\sf{ \int \frac{ \{x(\pi + 49) \} ^{ \frac{15}{7} } }{ {\pi}^{2}( {x}^{\pi} + 7) } \: dx }}} \: \: \bigstar \end{gathered}$

$\begin{gathered} \qquad \dashrightarrow\large{\sf{ \int \dfrac{ x^{ \frac{15}{7} } (\pi + 49) ^{ \frac{15}{7} } }{ {\pi}^{2}( {x}^{\pi} + 7) } \: \dfrac{dt}{\pi \: \displaystyle x ^{15 /7 }} }} \end{gathered}$

$\begin{gathered} \qquad \dashrightarrow\large{\sf{ \dfrac{ { \ } (\pi + 49) ^{ \frac{15}{7} } }{ {\pi}^{3}}\int\dfrac{1}{t} \: dt }} \end{gathered}$

$\begin{gathered} \qquad \dashrightarrow\large{\sf{ \dfrac{{ \ }(\pi + 49) ^{ \frac{15}{7}} } {{\pi}^{3}}\: log \: t \: + \: C}}\end{gathered}$

$\begin{gathered} \qquad \dashrightarrow\large{\sf{ \dfrac{{ \ } (\pi + 49) ^{ \frac{15}{7}} \displaystyle \: log( {x}^{ \pi} \: + \: 7 )}{ {\pi}^{3}}\: + \: C}} \end{gathered}$

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Additional information : -

$\begin{gathered}\begin{gathered}\: \: \: \: \: \: \begin{gathered}\begin{gathered} \footnotesize{\boxed{ \begin{array}{cc} \small\underline{\frak{\pmb{ \red{More \: Formulae}}}} \\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ {a}^{2} - {x}^{2} } = \dfrac{1}{2a} log\bigg | \dfrac{a + x}{a - x}\bigg | + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {x}^{2} - {a}^{2} } } = log |x + \sqrt{ {x}^{2} - {a}^{2} } | + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {a}^{2} - {x}^{2} } } = {sin}^{ - 1} \frac{x}{a} + c }\\ \\ \bigstar \: \bf{\displaystyle\int\sf \frac{dx}{ \sqrt{ {x}^{2} + {a}^{2} } } = log |x + \sqrt{ {x}^{2} + {a}^{2}} | + c}\\ \\ \end{array} }}\end{gathered}\end{gathered}\end{gathered}\end{gathered}$