HOW DO I DO THIS QUESTION: Integration (1 Viewer)

Denki · Apr 25, 2018

So recently while reading BNHA Manga, I realise this question:

Integration of (0 to ln(1+sqrt2))((e^x-e^-x)/2)^3 . ((e^x+e^-x)/2)^11

No clue pls help! Amswer 107/24
Thank you in advance

InteGrand · Apr 25, 2018

Denki said:
So recently while reading BNHA Manga, I realise this question:

Integration of (0 to ln(1+sqrt2))((e^x-e^-x)/2)^3 . ((e^x+e^-x)/2)^11

No clue pls help! Amswer 107/24
Thank you in advance

$\noindent I'll give you an outline of how to compute the \emph{in}definite integral $I = \int \left(\frac{e^{x} - e^{-x}}{2}\right)^{3} \left(\frac{e^{x} + e^{-x}}{2}\right)^{11} \, dx$. If you can do this, you should be able to do your question.$

$\noindent Write $\color{blue}\mathcal{S} = \frac{e^{x} - e^{-x}}{2}\color{black}$ and $\color{blue}\mathcal{C} = \frac{e^{x} + e^{-x}}{2}\color{black}$. Then $I = \int \mathcal{S}^{3} \mathcal{C}^{11}\, dx$. Write $I = \int \mathcal{S}^{2} \mathcal{C}^{11}\times \mathcal{S}\, dx$. Observe the following (easy-to-prove) facts:.$

$\noindent 1) $\mathcal{S}^{2} = \mathcal{C}^{2} - 1$
$\noindent 2) $\mathcal{C}' = \mathcal{S}.$

$\noindent Hence $I = \int \left(\mathcal{C}^{2} - 1\right) \mathcal{C}^{11}\, \mathcal{C}' \, dx$. Hence substituting $u = \mathcal{C}$ (so $du = \mathcal{C}'\, dx$), we have $I = \int \left(u^{2} - 1\right) u^{11}\, du = \cdots$. (Final answer for the indefinite integral should be re-expressed in terms of $x$, using the definition of $\mathcal{C}$.)$

InteGrand · Apr 25, 2018

HeroWise · Apr 25, 2018

LOL, Ths is from the januray issue. i remeber i was tryna figure it out too LOL
i asked my teacher and he did it fir me

Carrying on wat integrand said,

sinh^-1x is ln(1+sqrt(x^2+1))

if u put 1 as x then its the same as ln|1+sqrt(2)|

and sinh(sin^-1(1))=1

simplifies it a lot

if u need the wrking out i can do it fir u

HOW DO I DO THIS QUESTION: Integration (1 Viewer)

Denki

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