Let $$f(x)$$ be a polynomial of degree three such that $$f(0)=1$$,$$f(1)=2$$ and $$0$$ is a critical point of $$f(x)such$$ that $$f(x)$$ does not have a local extremum at $$0$$. Then∫$$f(x)x2+1dx$$ is equal to

Question

Let $$f(x)$$ be a polynomial of degree three such that $$f(0)=1$$,$$f(1)=2$$ and $$0$$ is a critical point of $$f(x)such$$ that $$f(x)$$ does not have a local extremum at $$0$$. Then∫$$f(x)x2+1dx$$ is equal to

Infinity Learn · Accepted Answer

Let $$f(x)=ax3+bx2+cx+d$$. $$Sincef(0)=1$$ so $$d=1$$. Moreover,  $$f(1)=2a$$ $$+$$ b $$+$$ c $$=$$ $$1$$ since $$0$$ is a critical point, $$0$$ $$=f$$′(0)=3a.0+2b.0+C$$ .i.e.,  $$C=0$$.  Also  f′′$$(x)=6ax+2band$$ $$sincef(x) does not have extremum at $$0$$ so $$0$$ $$=f$$′′$$(0)=b$$.Hence a $$=$$ $$1$$  and ∫$$f(x)x2+1dx=$$∫$$x3+1x2+1dx=$$∫x−$$122xx2+1+1x2+1dx=12x2$$−log⁡$$x2+1+$$$$tan$$−$$1$$⁡$$x+C$$.

Let $f (x)$ be a polynomial of degree three
such that $f (0) = 1, f (1) = 2 and 0$ is a critical point of $f (x)$
such that $f (x)$ does not have a local extremum at 0. Then
$\int \frac{f (x)}{x^{2} + 1} d x$ is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

Let f(x) be a polynomial of degree three such that f(0)=1,f(1)=2 and 0 is a critical point of f(x)such that f(x) does not have a local extremum at 0. Then∫f(x)x2+1dx is equal to

Remember concepts with our Masterclasses.

Ready to Test Your Skills?

free study material

Let $f (x)$ be a polynomial of degree three
such that $f (0) = 1, f (1) = 2 and 0$ is a critical point of $f (x)$
such that $f (x)$ does not have a local extremum at 0. Then
$\int \frac{f (x)}{x^{2} + 1} d x$ is equal to