Key Properties of Special Integrals for 12th Grade Calculus

Here are some important types of integrals that you should know for your calculus studies:

1. Gaussian Integral:

   • The integral $I = \int_{-\infty}^{\infty} e^{-x^2} \, dx = \sqrt{\pi}$ is really important in fields like probability and statistics.

2. Exponential Integral:

   • The integral $\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C$ is often used when solving equations that involve rates of change.

3. Trigonometric Integrals:

   • These integrals help us with problems that involve waves or oscillations. For example:
     • $\int \sin(x) \, dx = -\cos(x) + C$
     • $\int \cos(x) \, dx = \sin(x) + C$

4. Area Under the Curve:

   • The definite integral $A = \int_{a}^{b} f(x) \, dx$ helps you find the area between a curve and the x-axis from $x=a$ to $x=b$.

5. Integration by Parts:

   • This is a method you can use. It's based on the product rule and is written as: $\int u \, dv = uv - \int v \, du$

Understanding these properties will help you solve more advanced integrals with confidence!