Integrate the following :
(i) `int(t-(1)/(t))^(2)" dt "`
(ii)`intsin(10t-50)" dt "`
(iii)`inte^((100t+6))" dt "`

Answer

Let's solve the given integrals step by step.

### (i) Integrate \( \int \left(t - \frac{1}{t}\right)^2 dt \)

1. **Expand the integrand**:
\[ \left(t - \frac{1}{t}\right)^2 = t^2 - 2t\left(\frac{1}{t}\right) + \left(\frac{1}{t}\right)^2 = t^2 - 2 + \frac{1}{t^2} \]

2. **Rewrite the integral**:
\[ \int \left(t^2 - 2 + \frac{1}{t^2}\right) dt = \int t^2 dt - \int 2 dt + \int \frac{1}{t^2} dt \]

3. **Integrate each term**:
- \( \int t^2 dt = \frac{t^3}{3} \)
- \( \int 2 dt = 2t \)
- \( \int \frac{1}{t^2} dt = -\frac{1}{t} \)

4. **Combine the results**:
\[ \int \left(t - \frac{1}{t}\right)^2 dt = \frac{t^3}{3} - 2t - \frac{1}{t} + C \]

### (ii) Integrate \( \int \sin(10t - 50) dt \)

1. **Use the integration formula for sine**:
\[ \int \sin(kx) dx = -\frac{1}{k} \cos(kx) + C \]

2. **Identify \( k \)**:
Here, \( k = 10 \).

3. **Integrate**:
\[ \int \sin(10t - 50) dt = -\frac{1}{10} \cos(10t - 50) + C \]

### (iii) Integrate \( \int e^{(100t + 6)} dt \)

1. **Use the integration formula for exponential functions**:
\[ \int e^{ax} dx = \frac{1}{a} e^{ax} + C \]

2. **Identify \( a \)**:
Here, \( a = 100 \).

3. **Integrate**:
\[ \int e^{(100t + 6)} dt = \frac{1}{100} e^{(100t + 6)} + C \]

### Final Answers:

1. \( \int \left(t - \frac{1}{t}\right)^2 dt = \frac{t^3}{3} - 2t - \frac{1}{t} + C \)
2. \( \int \sin(10t - 50) dt = -\frac{1}{10} \cos(10t - 50) + C \)
3. \( \int e^{(100t + 6)} dt = \frac{1}{100} e^{(100t + 6)} + C \)