The value of for which 4 x^3 + 4^x 4^x + x^4 , ,dx = ( 4^x + x^4 … - Vidyadip

MCQ

The value of $\lambda$ for which $\int {\frac{{4{x^3} + \lambda {4^x}}}{{{4^x} + {x^4}}}} \,\,dx = \log ({4^x} + {x^4}) + c$ is

A
$1$
✓
$log_e4$
C
$log_4e$
D
$4$

✓

Answer

Correct option: B.

$log_e4$

b
Put $4^{x}+x^{4}=t$

$\left(4^{x} \ln 4+4 x^{3}\right) d x=d t$

$\int {\frac{{{\rm{dt}}}}{{\rm{t}}}} = \ln {\rm{t}} + {\rm{c}}$

$ = \ln \left| {{4^x} + {{\rm{x}}^4}} \right| + {\rm{c}}$

$\therefore \lambda=\ln 4$

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