Evaluate the following integrals

• (i) ∫_R (G M m) / (x²) d x
• (ii) ∫_r2 (k q₁ q₂) / (x²) d x
• (iii) ∫_vu M v d v
• (iv) ∫₀^∞ x⁻¹ d x
• (v) ∫₀^π/2 sin x d x
• (vi) ∫₀^π/2 cos x d x
• (vii) ∫_−π/2^π/2 cos x d x

Let's evaluate each integral step by step.

Integral Evaluations

(i) ∫ R (G M m) / (x²) dx

This integral can be expressed as:

∫ (G M m) / (x²) dx = - (G M m) / x + C

(ii) ∫ r² (k q₁ q₂) / (x²) dx

For this integral, we have:

∫ (k q₁ q₂ r²) / (x²) dx = - (k q₁ q₂ r²) / x + C

(iii) ∫ v u M v dv

This integral simplifies to:

∫ u M v² dv = (u M / 3) v³ + C

(iv) ∫₀∞ x⁻¹ dx

This integral diverges, meaning it does not converge to a finite value.

(v) ∫₀^(π/2) sin x dx

The result of this integral is:

-cos x |₀^(π/2) = 1

(vi) ∫₀^(π/2) cos x dx

This integral evaluates to:

sin x |₀^(π/2) = 1

(vii) ∫₋(π/2)^(π/2) cos x dx

For this integral, we find:

sin x |₋(π/2)^(π/2) = 2

These evaluations cover the integrals you provided. Each integral has been calculated based on standard integration techniques.