Let $f(x)$ be a continuous function on the real line such that for any $x$, $\int_{0}^{x} f(t)dt = x^2(1+x^2)$. Then $f(2)$ is _________.
We are given a function $\int_{0}^{x} f(t)dt = x^2(1+x^2)$. To find $f(x)$, we differentiate both sides with respect to $x$. This follows from the Fundamental Theorem of Calculus.
Differentiate: $$\frac{d}{dx}\left(\int_{0}^{x} f(t) dt\right) = \frac{d}{dx}[x^2(1+x^2)].$$
By the Fundamental Theorem of Calculus, the left side becomes $f(x)$. Differentiate the right side with respect to $x$:
$$f(x) = \frac{d}{dx}[x^2(1+x^2)] = \frac{d}{dx}[x^2 + x^4].$$
Breaking it down: $$\frac{d}{dx}[x^2] = 2x$$ and $$\frac{d}{dx}[x^4] = 4x^3.$$
Therefore, $$f(x) = 2x + 4x^3.$$
Substitute $x=2$ to find $f(2)$: $$f(2) = 2(2) + 4(2)^3 = 4 + 4 \cdot 8 = 4 + 32 = 36.$$
Thus, $f(2) = 36$.
Upon comparison, the provided range was incorrectly stated as a minimum and maximum. Our computed solution $36$ does not fit "5,5". The correct range or comparison might have been incorrectly interpreted.
Important Questions from Definite Integrals
The Legendre polynomials P n(x), n = 0, 1, 2, ..., satisfying the orthogonailty condition \(\int_{{\rm{ - 1}}}^{\rm{1}} {{{\rm{P}}_{\rm{n}}}\left( {\rm{x}} \right){{\rm{P}}_{\rm{m}}}} \left( {\rm{x}} \right){\rm{dx}}\,{\rm{ = }}\,\frac{{\rm{2}}}{{{\rm{2n + 1}}}}{{\rm{\delta }}_{{\rm{nm}}}}\) on the interval [-1, +1], may be defined by the Rodrigues formula P n(x) = \(\frac{{\rm{1}}}{{{{\rm{2}}^{\rm{n}}}{\rm{n!}}}}\frac{{{{\rm{d}}^{\rm{n}}}}}{{{\rm{d}}{{\rm{x}}^{\rm{n}}}}}{\left( {{{\rm{x}}^{\rm{2}}}{\rm{ - 1}}} \right)^{\rm{n}}}\) . The value of the definite integral \(\int_{{\rm{ - 1}}}^{\rm{1}} {\left( {{\rm{4 + 2x - 3}}{{\rm{x}}^{\rm{2}}}{\rm{ + 4}}{{\rm{x}}^{\rm{3}}}} \right){{\rm{P}}_{\rm{3}}}\left( {\rm{x}} \right){\rm{dx}}} \) is
A parametric curve is defined \(x = cos\left(\frac{\Pi t}{2}\right) , Y= sin\left(\frac{\Pi t}{2}\right)\) in the range of \(0\leq t\leq 1\) . It is rotated about X-axis by 360°.
‘What is the area of the surface generated?
if \(\displaystyle\int\dfrac{\sin x}{\sin (x-a)}dx=Ax+B\log |sin(x-a)|+ C\) where A, B and c are real constants then:
Which of the following is NOT a property of definite integral?
The value of integral \(x\over cos^2 x\) is equal to
