The integral e ^ 3 _ e 2 x +5 e ^ 2 _ e 2 x e ^ 4 _ e x +5 e ^ 3 … - Vidyadip

MCQ

The integral $\int \frac{ e ^{3 \log _{e} 2 x }+5 e ^{2 \log _{ e } 2 x }}{ e ^{4 \log _{e} x }+5 e ^{3 \log _{e} x }-7 e ^{2 \log _{e} x }} dx , x > 0$, is equal to ....... .

(where $c$ is a constant of integration)

A
$\log _{ e }\left| x ^{2}+5 x -7\right|+ c$
✓
$4 \log _{ e }\left| x ^{2}+5 x -7\right|+ c$
C
$\frac{1}{4} \log _{ e }\left| x ^{2}+5 x -7\right|+ c$
D
$\log _{ e } \sqrt{ x ^{2}+5 x -7}+ c$

✓

Answer

Correct option: B.

$4 \log _{ e }\left| x ^{2}+5 x -7\right|+ c$

b
$\int \frac{ e ^{3 \log _{ e } 2 x }+5 e ^{2 \log _{ e } 2 x }}{ e ^{4 \log _{ e } x }+5 e ^{3 \log _{ e } x }-7 e ^{2 \log _{ e } x }} dx , x > 0$

$=\int \frac{(2 x )^{3}+5(2 x )^{2}}{ x ^{4}+5 x ^{3}-7 x ^{2}} d x =\int \frac{4 x ^{2}(2 x +5)}{ x ^{2}\left( x ^{2}+5 x -7\right)} d x$

$=4 \int \frac{ d \left( x ^{2}+5 x -7\right)}{\left( x ^{2}+5 x -7\right)}=4 \log _{ e }\left| x ^{2}+5 x -7\right|+ c$

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