Evaluate the definite integral:
int_{ 2 } ^ { 7 } (10 x + 5) dx =
To evaluate the definite integral ∫(10x + 5)dx from 2 to 7, you can follow these steps:
Step 1: Use the power rule for integration.
The power rule states that integrating a term of the form ax^n with respect to x gives you (a/(n+1))x^(n+1) + C, where C is the constant of integration.
Step 2: Apply the power rule to each term in the integrand.
For the integral of 10x, we have:
∫(10x)dx = (10/2)x^2 + C = 5x^2 + C
For the integral of 5, we have:
∫5dx = 5x + C
Step 3: Use the definite integral formula.
The definite integral from a to b of a function f(x) is given by:
∫[a, b] f(x)dx = F(b) - F(a), where F(x) is the antiderivative of f(x).
Step 4: Evaluate the definite integral.
Applying the definite integral formula to our integral from 2 to 7, we get:
∫[2, 7] (10x + 5)dx = (5x^2 + C) (evaluated from 2 to 7) - (5x + C) (evaluated from 2 to 7)
Evaluating at x = 7:
(5(7)^2 + C) - (5(7) + C)
Evaluating at x = 2:
(5(2)^2 + C) - (5(2) + C)
Finally, subtract the evaluation at x = 2 from the evaluation at x = 7:
[(5(7)^2 + C) - (5(7) + C)] - [(5(2)^2 + C) - (5(2) + C)]
Now you can simplify the expression to find the value of the definite integral.