Mathematics & Numerical Analysis

Using the additive property of definite integrals, split the integral at $x = 2$:
$\int_0^6 f(x)\,dx = \int_0^2 x^2\,dx + \int_2^6 (3x - 2)\,dx$.
Evaluate the first integral:
$\int_0^2 x^2\,dx = \left[\frac{x^3}{3}\right]_0^2 = \frac{8}{3} - 0 = \frac{8}{3}$.
Evaluate the second integral:
$\int_2^6 (3x - 2)\,dx = \left[\frac{3x^2}{2} - 2x\right]_2^6 = \left(\frac{3(36)}{2} - 2(6)\right) - \left(\frac{3(4)}{2} - 2(2)\right) = (54 - 12) - (6 - 4) = 42 - 2 = 40$.
Adding both parts:
$\frac{8}{3} + 40 = \frac{8 + 120}{3} = \frac{128}{3}$.

Detailed analysis of options:

• is incorrect.
• 0 is incorrect.
• $1/3$ is incorrect.
• $\frac{128}{3}$ is the exact and correct value.

By Euler's integral of the first kind in terms of Gamma functions:
$\int_0^{\pi/2} \sin^p x \cos^q x\,dx = \frac{\Gamma\left(\frac{p+1}{2}\right)\Gamma\left(\frac{q+1}{2}\right)}{2\Gamma\left(\frac{p+q+2}{2}\right)}$.
Setting $p = n$ and $q = 0$:
$\int_0^{\pi/2} \sin^n x\,dx = \frac{\Gamma\left(\frac{n+1}{2}\right)\Gamma\left(\frac{1}{2}\right)}{2\Gamma\left(\frac{n+2}{2}\right)}$.
Since $\Gamma(1/2) = \sqrt{\pi}$, this simplifies to:
$\frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{n+2}{2}\right)} \frac{\sqrt{\pi}}{2}$.

Detailed analysis of options:

• $\frac{\Gamma\left(\frac{n+1}{2}\right)}{\Gamma\left(\frac{n+2}{2}\right)}\frac{\sqrt{\pi}}{2}$ is the exact formula.
• Has $n/2$ and $(n+1)/2$ instead of $(n+1)/2$ and $(n+2)/2$.
• Inverts the ratio of Gamma arguments.
• Uses integer arguments $n$ and $n+1$.

Integration by parts formula: $\int u v' dx = u v - \int u' v dx$.
Let $u = \tan^{-1}x \implies u' = \frac{1}{1 + x^2}$.
Let $v' = x \implies v = \frac{x^2}{2}$.
$\int x\tan^{-1}x \, dx = \frac{x^2}{2}\tan^{-1}x - \int \frac{x^2}{2(1 + x^2)} \, dx$
$= \frac{x^2}{2}\tan^{-1}x - \frac{1}{2}\int \frac{(x^2 + 1) - 1}{1 + x^2} \, dx$
$= \frac{x^2}{2}\tan^{-1}x - \frac{1}{2}\int \left(1 - \frac{1}{1 + x^2}\right) dx$
$= \frac{x^2}{2}\tan^{-1}x - \frac{1}{2}x + \frac{1}{2}\tan^{-1}x + C$.

Option breakdown:

• Uses $\cot^{-1}x$ instead of $\tan^{-1}x$.
• Uses $\sec^{-1}x$ instead of $\tan^{-1}x$.
• Exactly matches the computed antiderivative.
• Has incorrect coefficients $-1/4$ and $-1/2$.

Evaluation of definite integral using double-angle identity:
Note that $\sin\frac{x}{2} \cos\frac{x}{2} = \frac{1}{2}\sin x$.