If -x3x4-3x2-2dx-ln-x2-2-fx-C- where C is constant of integration- then the value of f7 is

Let I=∫x^3x^4+3x^2+2dx Put x^2=t⇒ 2xdx=dt ∴ I=12∫tt^2+3t+2dt ⇒ I=12∫(2t+2 1t+1)dt ⇒ I=ln|x^2+2| 12ln|x^2+1|+C ⇒ I= ln|x^2+2√(x^2+1)|+C ∴ f(x)=x^2+1 Hence, f(7)= ...

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Byju's Answer

Standard XII

Mathematics

Properties of Iota

If ∫x3x4+3x2+...

Question

If $\int \frac{x^{3}}{x^{4} + 3 x^{2} + 2} d x = ln ∣ ∣ ∣ ∣ \frac{x^{2} + 2}{\sqrt{f (x)}} ∣ ∣ ∣ ∣ + C,$ where $C$ is constant of integration, then the value of $f (7)$ is

5 \sqrt{2}

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50

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29

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15

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Solution

The correct option is B $50$
Let $I = \int \frac{x^{3}}{x^{4} + 3 x^{2} + 2} d x$
Put $x^{2} = t \Rightarrow 2 x d x = d t$
$∴ I = \int \frac{t}{t^{2} + 3 t + 2} d t$
$\Rightarrow I = \frac{1}{2} \int (\frac{2}{t + 2} - \frac{1}{t + 1}) d t$
$\Rightarrow I = ln | x^{2} + 2 | - \frac{1}{2} ln | x^{2} + 1 | + C$
$\Rightarrow I = ln ∣ ∣ ∣ \frac{x^{2} + 2}{\sqrt{x^{2} + 1}} ∣ ∣ ∣ + C$
$∴ f (x) = x^{2} + 1$
Hence, $f (7) = 50$

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If ∫x3x4+3x2+2dx=ln∣∣ ∣∣x2+2√f(x)∣∣ ∣∣+C, where C is constant of integration, then the value of f(7) is

The correct option is B 50Let I=∫x3x4+3x2+2dx Put x2=t⇒2xdx=dt ∴I=∫tt2+3t+2dt ⇒I=12∫(2t+2−1t+1)dt ⇒I=ln|x2+2|−12ln|x2+1|+C ⇒I=ln∣∣∣x2+2√x2+1∣∣∣+C ∴f(x)=x2+1 Hence, f(7)=50

If $\int \frac{x^{3}}{x^{4} + 3 x^{2} + 2} d x = ln ∣ ∣ ∣ ∣ \frac{x^{2} + 2}{\sqrt{f (x)}} ∣ ∣ ∣ ∣ + C,$ where $C$ is constant of integration, then the value of $f (7)$ is