Let lim x - 0lna ex-sin b x-x-2 - If I -b-3a2-3 x-4 x2-2 -1-x-x2 d x- then I2 equals

Lim_x→0ln(ae^x+sin bx)x = nbsp; lim_x→0ln[(ae^x+sin bx 1)+1](ae^x+sin bx 1)×ae^x+sin bx 1x ⇒lim_x→0ae^x+sin bx 1x = 2 ⇒ a=1,b= 3 Now, nbsp; I= ∫_b+3^a2+3x+4 ...

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Byju's Answer

Standard XII

Mathematics

Integration by Substitution

Let lim x → 0...

Question

Let $lim x \to 0 \frac{ln (a e^{x} + sin b x)}{x} = - 2 .$ If $I = a \int b + 3 \frac{2 + 3 x + 4 x^{2}}{2 \sqrt{1 + x + x^{2}}} d x,$ then $I^{2}$ equals

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Solution

$lim x \to 0 \frac{ln (a e^{x} + sin b x)}{x}$
$= lim x \to 0 \frac{ln [(a e^{x} + sin b x - 1) + 1]}{(a e^{x} + sin b x - 1)} \times \frac{a e^{x} + sin b x - 1}{x}$ $\Rightarrow lim x \to 0 \frac{a e^{x} + sin b x - 1}{x} = - 2 \Rightarrow a = 1, b = - 3$

Now, $I = a \int b + 3 \frac{2 + 3 x + 4 x^{2}}{2 \sqrt{1 + x + x^{2}}} d x$
Multiply numerator and denominator by $x$
$I = 1 \int 0 \frac{2 x + 3 x^{2} + 4 x^{3}}{2 \sqrt{x^{2} + x^{3} + x^{4}}} d x$
Put $x^{2} + x^{3} + x^{4} = t^{2}$ $I = \sqrt{3} \int 0 \frac{2 t d t}{2 t} = \sqrt{3}$
$\Rightarrow I^{2} = 3$

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Let limx→0ln(aex+sinbx)x=−2. If I=a∫b+32+3x+4x22√1+x+x2dx, then I2 equals

limx→0ln(aex+sinbx)x =limx→0ln[(aex+sinbx−1)+1](aex+sinbx−1)×aex+sinbx−1x ⇒limx→0aex+sinbx−1x=−2⇒a=1,b=−3 Now, I=a∫b+32+3x+4x22√1+x+x2dx Multiply numerator and denominator by x I=1∫02x+3x2+4x32√x2+x3+x4dx Put x2+x3+x4=t2 I=√3∫02tdt2t=√3 ⇒I2=3

Let $lim x \to 0 \frac{ln (a e^{x} + sin b x)}{x} = - 2 .$ If $I = a \int b + 3 \frac{2 + 3 x + 4 x^{2}}{2 \sqrt{1 + x + x^{2}}} d x,$ then $I^{2}$ equals