Calculus A, Evaluate the following integral : �ç (x^2+3x-2)dx
intrgral of x^n dx = (1/n+1) x^(n+1)
(1/3)x^3 + (3/2)x^2 - 2 x
To evaluate the integral of \(x^2 + 3x - 2\), we can use the power rule and linear rule of integration. Here's how to do it step by step:
Step 1: Separate the terms of the integrand.
The integral of a sum of functions is equal to the sum of the integrals of each individual function. So, we can break down the integral as follows:
\(\int (x^2 + 3x - 2) \, dx = \int x^2 \, dx + \int 3x \, dx - \int 2 \, dx\)
Step 2: Apply the power rule of integration.
The power rule of integration states that \(\int x^n \, dx = \frac{{x^{n+1}}}{{n+1}} + C\), where \(n\) is any real number (excluding -1).
Applying the power rule to each term, we get:
\(\int x^2 \, dx = \frac{{x^3}}{3} + C_1\)
\(\int 3x \, dx = \frac{{3x^2}}{2} + C_2\)
\(\int 2 \, dx = 2x + C_3\)
Step 3: Combine the results.
Combining the integrals, we have:
\(\int (x^2 + 3x - 2) \, dx = \frac{{x^3}}{3} + C_1 + \frac{{3x^2}}{2} + C_2 - 2x + C_3\)
We can simplify this result:
\(\int (x^2 + 3x - 2) \, dx = \frac{{x^3}}{3} + \frac{{3x^2}}{2} - 2x + C\)
where \(C = C_1 + C_2 + C_3\) represents the constant of integration.