Unit 3: Addition, Subtraction, Constant Multiplication of Indefinite integrals

When dealing with indefinite integrals, we encounter three fundamental operations: addition, subtraction, and constant multiplication. These operations allow us to simplify expressions and solve problems effectively in calculus.

• Addition: [latex]\int (f(x) + g(x)) \, dx = \int f(x) \, dx + \int g(x) \, dx[/latex]
• Subtraction: [latex]\int (f(x) - g(x)) \, dx = \int f(x) \, dx - \int g(x) \, dx[/latex]
• Constant multiplication: [latex]\int c\,f(x) \, dx = c\, \int f(x) \, dx[/latex] where [latex]c[/latex] is an constant

These operations provide valuable tools for manipulating and solving indefinite integrals efficiently, aiding in the breakdown of complex expressions and evaluation of integrals. Understanding these properties is crucial for mastering calculus concepts and applications.

The above operations also apply to definite integrals; for more details, please refer to Unit 8.

Practice Questions:

Evaluate the following indefinite integrals before referring to the video for comprehensive solutions.

Q1. [latex]\int 4^x + x^4 \; dx[/latex]

Q2. [latex]\int e^x + x^e + e \; dx[/latex]

Q3. [latex]\int (\frac{3}{2})^x - 2 x^{-\frac{3}{2}}\; dx[/latex]

Q4. [latex]\int \frac{4}{x} - \frac{5}{x^2} \;dx[/latex]

Q5.  [latex]\int \frac{\sqrt{x}}{5}- \frac{5}{\sqrt{x}}\; dx[/latex]

Q6.  [latex]\int x^{-2}(1-5x)\; dx[/latex]

Q7.  [latex]\int \frac{1+2t}{\sqrt{t}} \; dt[/latex]

Q8. [latex]\int \frac{x^5-2\sin x}{3}  \;dx[/latex]

Q9. [latex]\int 10\cos t  + 3\tan t \sec t \;dt[/latex]

Q10. [latex]\int 5\sec\theta \tan\theta - \frac{\sec^2 \theta}{2}  \;d\theta[/latex]