Interesting Integrals 3

Calculus Level 3

$\large\int_{1}^{7}\dfrac{(\ln{(x)}) ^4}{(\ln{(x)})^4+(\ln{(8-x)})^4}\,dx \,=\, ?$

The answer is 3.

2 solutions

Chew-Seong Cheong
Jul 9, 2016

$\begin{aligned} I & = \int_1^7 \frac {\ln^4 x}{\ln^4 x + \ln^4 (8-x)} dx & \small \color{#3D99F6}{\text{By }\int_a^b f(x) \ dx = \int_a^b f(a+b-x) \ dx} \\ & = \int_1^7 \frac {\ln^4 (8-x)}{\ln^4 (x-8) + \ln^4 x} dx \\ \implies 2I & = \int_1^7 \frac {\ln^4 x}{\ln^4 x + \ln^4 (8-x)} dx + \int_1^7 \frac {\ln^4 (8-x)}{\ln^4 (x-8) + \ln^4 x} dx \\ & = \int_1^7 \frac {\ln^4 x+ \ln^4 (8-x)}{\ln^4 x + \ln^4 (8-x)} dx \\ & = \int_1^7 1 \ dx = x \ \big|_1^7 = 6 \\ \implies I & = \boxed{3} \end{aligned}$

Krishna Shankar , natural log should be $\ln$ ( $\text{LN}$ ) and not $\In$ ( $\text{IN}$ ), put a backslash "\" before all functions in LaTex including ln, sin, cos, tan, etc ( $\ln, \sin, \cos, \tan,$ ). See the backslash does not work with $\text{In}$ (in red above) because it is not a function. You can put your mouse cursor on top of the formulas to see the LaTex codes used.

Chew-Seong Cheong - 4 years, 11 months ago

Aditya Gahlawat
Jul 9, 2016

Use a+b-x property where a and b are the lower and upper limits of the definite integral respectively

Interesting Integrals 3

The answer is 3.

2 solutions

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