Let f be a polynomial function such that for all real $x, f (x^{2} + 1) = x^{4} + 5 x^{2} + 2$ , then $\int f (x) d x$ is

\frac{x^{3}}{3} + \frac{3 x^{2}}{2} - 2 x + c

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\frac{x^{3}}{3} + \frac{3 x^{2}}{2} + 2 x + c

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\frac{x^{3}}{3} - \frac{3 x^{2}}{2} - 2 x + c

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\frac{x^{3}}{3} - \frac{3 x^{2}}{2} + 2 x + c

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Solution

The correct option is A $\frac{x^{3}}{3} + \frac{3 x^{2}}{2} - 2 x + c$
$f (x^{2} + 1) = x^{4} + 5 x^{2} + 2$

$f (x^{2} + 1) = x^{4} + 2 x^{2} + 1 + 3 x^{2} + 3 - 1$

$f (x^{2} + 1) = {(x^{2} + 1)}^{2} + 3 (x^{2} + 1) - 2$

$f (x) = x^{2} + 3 x - 2$

Let $I = \int f (x) d x$

$= \int (x^{2} + 3 x - 2) d x$

$I = \frac{x^{3}}{3} + \frac{3 x^{3}}{3} - 2 x + c$

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Similar questions

\frac{x^{3}}{3} - 2 \sqrt{x} + \frac{5}{x^{2}}

Δ_{1} = ∣ ∣ ∣ ∣ \begin{matrix} x & 3 & 3 - 2 & x & 3 - 2 & - 2 & x \end{matrix} ∣ ∣ ∣ ∣

Δ_{2} = ∣ ∣ ∣ \begin{matrix} x & 3 - 2 & x \end{matrix} ∣ ∣ ∣

\int \frac{x^{3}}{x + 1} d x

(a) x + \frac{x^{2}}{2} + \frac{x^{3}}{3} - \log |1 - x| + C (b) x + \frac{x^{2}}{2} - \frac{x^{3}}{3} - \log |1 - x| + C (c) x - \frac{x^{2}}{2} - \frac{x^{3}}{3} - \log |1 + x| + C (d) x - \frac{x^{2}}{2} + \frac{x^{3}}{3} - \log |1 + x| + C

\int \frac{x^{5}}{\sqrt{1 + x^{3}}} d x =

2 x^{2} + x - 1 = 0

2 x - \frac{1}{x} + {(4 x^{2} + \frac{1}{x^{2}})}^{2} + {(8 x^{3} - \frac{1}{x^{3}})}^{3}

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