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Help me solve the control problem urgently​

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1. My Russian is a bit rusty. I think you're asking to find the antiderivatives in the first part:

а) $$\displaystyle \int 2x^n - 5x \, dx = \boxed{\frac2{n+1} x^{n+1} - \frac52 x^2 + C}$$

This follows from the power rule for differentiation,

$$\dfrac d{dx} x^n = n x^{n-1}$$

I'm not sure what the power on the first term is, so I just use a general real number n. This solution is correct as long as n ≠ -1; otherwise we would have

$$\displaystyle \int 2x^{-1} - 5x = 2\ln|x| - \frac52 x^2 + C$$

б) Using the fact that

$$\dfrac d{dx} \sin(x) = \cos(x)$$

as well as the power rule from part (a),

$$\displaystyle \int 3 \cos(x) - x \, dx = \boxed{3 \sin(x) - \frac12 x^2 + C}$$

в) By the chain rule,

$$\dfrac d{dx} \sin(5x) = 5 \cos(5x)$$

$$\dfrac d{dx} \cos(3x) = -3 \sin(3x)$$

Hence

$$\displaystyle \int \cos(5x) - \frac16 \sin(3x) \, dx = \boxed{\frac15 \sin(5x) + \frac1{18} \cos(3x) + C}$$

2. These just look like standard definite integrals. Using the known derivatives mentioned in part (1) in conjunction with the fundamental theorem of calculus, we have

a)

$$\displaystyle \int_0^1 x^2 \, dx = \frac13 x^3 \bigg|_{x=0}^{x=1} = \frac13 (1^3 - 0^3) = \boxed{\frac13}$$

б)

$$\displaystyle \int_0^{\frac\pi2} \sin(x) \, dx = -\cos(x) \bigg|_{x=0}^{x=\frac\pi2} = -\left(\cos\left(\frac\pi2\right) - \cos(0)\right) = \boxed{1}$$
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