Question

$\int\limits^5_2 2x/5x²+1 dx ,Evaluate the given integral expression:$

Answer 1

HELLO DEAR,

put (5x² + 1) = t
=> 10x.dx = dt
also, [x = 5 , t = 126] and [x = 2 , t = 21]

$\bold{\int\limits^{126}_{21} [1/5(dt/t)] }$

$\bold{1/5[log|t|]^{126}_{21}}$

$\bold{1/5[log126 - log21]}$

$\bold{log{126\over11}/5}$

$\bold{{log6\over5}}$



I HOPE ITS HELP YOU DEAR,
THANKS

Answer 2

we have to get the value of $\int\limits^5_2{\frac{2x}{5x^2+1}}\,dx$

Let f(x) = 5x² + 1
now differentiate with respect to x,
f'(x) = 10x , so we should arrange numerator in such a way that it contains 10x for getting integration form.

e.g., $\int\limits^5_2{\frac{1}{5}\frac{10x}{5x^2+1}}\,dx$

=$\frac{1}{5}\int\limits^5_2{\frac{10x}{5x^2+1}}\,dx$
now it seems like $\int{\frac{f'(x)}{f(x)}}\,dx$
we know, $\int{\frac{f(x)}{f(x)}}\,dx=logf(x)$

so, $\frac{1}{5}[log(5x^2+1)]^5_2$

=$\frac{1}{5}[log(126) - log(21)]$

=$\frac{1}{5}log\left(\frac{126}{21}\right)$

=$\frac{1}{5}log6$