Question

choose the correct answer; integral of [10x^9 + 10^xlog_{e^10x}]/(10^x + x^10).dx  (A).(10^x - x^10) + c ( B ). (10^x + x^10) + c      (C). (10^x - x^10)^{-1} + c     (D).   log(10^x + x^10) + c

Answer 1

question is $\bf{\int{\frac{10x^9+10^x.log_e^{10}}{10^x+x^{10}}}\,dx}$

here, Let f(x) = 10^x + x^10
differentiate with respect to x
f'(x) = 10^x.log10 + 10x^9
hence, it is clear that given integration is like as
$\bf{\int{\frac{f'(x)}{f(x)}}\,dx}$

but we know, according to rule of integration,
if $\bf{\int{\frac{f'(x)}{f(x)}}\,dx}$ is given
then, integration will be $\bf{logf(x)+C}$
so, $\bf{\int{\frac{10x^9+10^x.log_e^{10}}{10^x+x^{10}}}\,dx=log(10^x+x^{10})+C}$

hence, option (D) is correct.

Answer 2

HELLO DEAR,

given :- $\bold{\int{\frac{10x^9 + 10^x log_{e^{10x}}}{10^x + x^10)}}.dx}$

let t = $\bold{10^x + x^10}$

$\bold{\Rightarrow dt/dx = 10^xlog^{10x} + 10x^9}$

$\bold\Rightarrow {dx = dt/(10^xlog_{e^{10x}} + 10x^9)}$

so, $\bold{\int{\frac{10x^9 + 10^xlog_{e^{10x}}}{10^x + x^10)}.dt/(10^xlog_{e^{10x}} + 10x^9)}}$

$\Rightarrow \bold{int{1/t}\,dt}$

$\Rightarrow \bold{log|t| + c}$

put the value of t in above function.

$\huge{\bold{\Rightarrow \log|10^x + x^{10}| + c} }$

I HOPE ITS HELP YOU DEAR,
THANKS