$\int (brain ) +(brain)^2d(brain)$

Calculus Level 5

Given

$I_{a}=\int_{1.0002014}^{e}\left[\frac{(\cos(a\ln x)(1-x\ln x)}{(x(\ln x)^{2})}+\frac{a\sin a\ln x}{x\ln x}\right]\exp x dx$

and

$I_{10^{7}}-I_{0} =\frac{\exp b\cos h}{i}-\frac{\exp d}{f}[1-\cos10^{7}] -\frac{\exp b}{c},$

find the value of

$\left[\frac{h(b-c)}{df}\right].$

Note: [.] is the greatest integer function.

The answer is 740.

2 solutions

Milun Moghe
Dec 30, 2013

the jist of the problem is to think about a as an independent parameter .If we differenciate the integral with respect to parameter a we with a simplified form of the integral

$\frac{dI_{a}}{da} =\int_{1.0002014}^{e}e^{x}\frac{d}{da}[\frac{cos(alnx)(1-xlnx)}{x(lnx)^{2}}+\frac{asin(alnx)}{x(lnx)}]dx$

$=\int_{1.0002014}^{e}e^{x}[sin(alnx)+\frac{acos(alnx)}{x}]dx$

$=e^{e}sin(a)-e^{1.0002014}sin(aln(1.0002014)$

$=e^{e}sina-e^{1.0002014}sin(0.0002014a)$

$I_{a}=\frac{e^{1.0002014}cos1.0002014}{0.0002014}-e^{e}cosa + c$

we can the put a=10000000 and 0 to ge the required value b=1.0002014, h=2014,i=0.0002014=c,lnd=1,f=1,

final answer is 740

Title should be " Brilliant and wolfram-alpha go hand in hand " :P

Deepanshu Gupta - 6 years, 2 months ago

Shouvik Das
Mar 29, 2014

Awesome question. Really feeling happy by solving it. Nothing new to say. Tried the same way given in other solution by Milun Moghe...Firstly tried to express it in the form of exp(x)[f(x)+fdash(x)]. then just integrating it to exp(x)*f(x)... After integrating got the function I(a) after putting the values i.e "e" and "1.0002014". Then just substituted the values of "a" by 10 to the power 7 and 0.. then got the answer as: "740.83461". After that [740.83461] = 740. Thus the final answer is "740"..........................

∫ ( b r a i n ) + ( b r a i n ) 2 d ( b r a i n ) \int (brain ) +(brain)^2d(brain) ∫ ( b r a i n ) + ( b r a i n ) 2 d ( b r a i n )

The answer is 740.

2 solutions

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$\int (brain ) +(brain)^2d(brain)$