Evaluate the following integrals
(i) `int_(R)^(oo)(GMm)/(x^(2))dx`
(ii) `int_(r_(1))^(r_(2)) -k(q_1q_2)/(x^(2))dx`
(iii) `int_(u)^(v) Mv dv`
(iv) `int_(0)^(oo) x^(-1//2) dx`
(v) `int_(0)^(pi//2) sinx dx `
(vi) `int_(0)^(pi//2) cosx dx`
(vii) `int_(-pi//2)^(pi//2) cos x dx`

Explanation

Let's evaluate the given integrals step by step.

### (i) Evaluate `∫_(R)^(∞) (GMm)/(x^2) dx`

1. **Identify constants**: Here, \( GMm \) is a constant.
2. **Factor out the constant**:
\[ \int_{R}^{\infty} \frac{GMm}{x^2} dx = GMm \int_{R}^{\infty} x^{-2} dx \]
3. **Integrate**: The integral of \( x^{-2} \) is:
\[ \int x^{-2} dx = -\frac{1}{x} + C \]
4. **Apply limits**:
\[ = GMm \left[-\frac{1}{x}\right]_{R}^{\infty} = GMm \left(0 + \frac{1}{R}\right) = \frac{GMm}{R} \]

**Final answer**: \( \frac{GMm}{R} \)

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### (ii) Evaluate `∫_(r1)^(r2) -k(q1q2)/(x^2) dx`

1. **Identify constants**: Here, \( -k(q_1q_2) \) is a constant.
2. **Factor out the constant**:
\[ -k(q_1q_2) \int_{r_1}^{r_2} x^{-2} dx \]
3. **Integrate**:
\[ = -k(q_1q_2) \left[-\frac{1}{x}\right]_{r_1}^{r_2} = -k(q_1q_2) \left(-\frac{1}{r_2} + \frac{1}{r_1}\right) \]
4. **Simplify**:
\[ = k(q_1q_2) \left(\frac{1}{r_1} - \frac{1}{r_2}\right) \]

**Final answer**: \( k(q_1q_2) \left(\frac{1}{r_1} - \frac{1}{r_2}\right) \)

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### (iii) Evaluate `∫_(u)^(v) Mv dv`

1. **Identify constants**: Here, \( M \) is a constant.
2. **Factor out the constant**:
\[ M \int_{u}^{v} v dv \]
3. **Integrate**:
\[ = M \left[\frac{v^2}{2}\right]_{u}^{v} = M \left(\frac{v^2}{2} - \frac{u^2}{2}\right) \]
4. **Simplify**:
\[ = \frac{M}{2} (v^2 - u^2) \]

**Final answer**: \( \frac{M}{2} (v^2 - u^2) \)

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### (iv) Evaluate `∫_(0)^(∞) x^(-1/2) dx`

1. **Integrate**:
\[ \int x^{-1/2} dx = 2x^{1/2} + C \]
2. **Apply limits**:
\[ = \left[2x^{1/2}\right]_{0}^{\infty} = 2(\infty - 0) = \infty \]

**Final answer**: \( \infty \)

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### (v) Evaluate `∫_(0)^(π/2) sin(x) dx`

1. **Integrate**:
\[ \int sin(x) dx = -cos(x) + C \]
2. **Apply limits**:
\[ = \left[-cos(x)\right]_{0}^{\frac{\pi}{2}} = -cos\left(\frac{\pi}{2}\right) + cos(0) = 0 + 1 = 1 \]

**Final answer**: \( 1 \)

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### (vi) Evaluate `∫_(0)^(π/2) cos(x) dx`

1. **Integrate**:
\[ \int cos(x) dx = sin(x) + C \]
2. **Apply limits**:
\[ = \left[sin(x)\right]_{0}^{\frac{\pi}{2}} = sin\left(\frac{\pi}{2}\right) - sin(0) = 1 - 0 = 1 \]

**Final answer**: \( 1 \)

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### (vii) Evaluate `∫_(-π/2)^(π/2) cos(x) dx`

1. **Integrate**:
\[ \int cos(x) dx = sin(x) + C \]
2. **Apply limits**:
\[ = \left[sin(x)\right]_{-\frac{\pi}{2}}^{\frac{\pi}{2}} = sin\left(\frac{\pi}{2}\right) - sin\left(-\frac{\pi}{2}\right) = 1 - (-1) = 2 \]

**Final answer**: \( 2 \)

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