The integral ∫ ( 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)

Question

The integral ∫ ( 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 | x 2+5 x -7|+ c (B) 4 log e|x2+5 x-7|+c (C) (1/4) log e | x 2+5 x -7|+ c (D)  log e √x2+5 x-7+c

Tardigrade Admin · Accepted Answer

∫ ( 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 =∫ ((2 x )3+5(2 x )2/ x 4+5 x 3-7 x 2) d x =∫ (4 x 2(2 x +5)/ x 2( x 2+5 x -7)) dx =4 ∫ ( d ( x 2+5 x -7)/( x 2+5 x -7))=4 log e | x 2+5 x -7|+ c

Q. The integral $\int \frac{e ^{3 l o g_{e} 2 x} + 5 e ^{2 l o g_{e} 2 x}}{e ^{4 l o g_{e} x} + 5 e ^{3 l o g_{e} x} - 7 e ^{2 l o g_{e} x}} d x, x > 0$ , is equal to :
(where $c$ is a constant of integration)

$lo g_{e} ∣ ∣ x^{2} + 5 x - 7 ∣ ∣ + c$

$4 lo g_{e} ∣ ∣ x^{2} + 5 x - 7 ∣ ∣ + c$

$\frac{1}{4} lo g_{e} ∣ ∣ x^{2} + 5 x - 7 ∣ ∣ + c$

$lo g_{e} x^{2} + 5 x - 7 + c$

Q. The integral ∫e4loge​x+5e3loge​x−7e2loge​xe3loge​2x+5e2loge​2x​dx,x>0, is equal to : (where c is a constant of integration)

loge​∣∣​x2+5x−7∣∣​+c

4loge​∣∣​x2+5x−7∣∣​+c

41​loge​∣∣​x2+5x−7∣∣​+c

loge​x2+5x−7​+c

∫e4loge​x+5e3loge​x−7e2loge​xe3loge​2x+5e2loge​2x​dx,x>0 =∫x4+5x3−7x2(2x)3+5(2x)2​dx=∫x2(x2+5x−7)4x2(2x+5)​dx =4∫(x2+5x−7)d(x2+5x−7)​=4loge​∣∣​x2+5x−7∣∣​+c

Q. The integral $\int \frac{e ^{3 l o g_{e} 2 x} + 5 e ^{2 l o g_{e} 2 x}}{e ^{4 l o g_{e} x} + 5 e ^{3 l o g_{e} x} - 7 e ^{2 l o g_{e} x}} d x, x > 0$ , is equal to :
(where $c$ is a constant of integration)

$lo g_{e} ∣ ∣ x^{2} + 5 x - 7 ∣ ∣ + c$

$4 lo g_{e} ∣ ∣ x^{2} + 5 x - 7 ∣ ∣ + c$

$\frac{1}{4} lo g_{e} ∣ ∣ x^{2} + 5 x - 7 ∣ ∣ + c$

$lo g_{e} x^{2} + 5 x - 7 + c$