Problem: Solve the definite integral ∫(4x^2 + 36x + 10) dx from a = 0 to b = 6.
Solution:
Using the power rule, we can integrate each term separately.
∫(4x^2 + 36x + 10) dx = (4/3)x^3 + (18)x^2 + (10x) + C
To find the definite integral from a = 0 to b = 6, we substitute the values into the antiderivative:
∫[0,6] (4x^2 + 36x + 10) dx = [(4/3)(6)^3 + (18)(6)^2 + (10)(6)] - [(4/3)(0)^3 + (18)(0)^2 + (10)(0)]
= (4/3)(216) + (18)(36) + (10)(6) - 0
= 288 + 648 + 60
= 996
Therefore, the value of the definite integral is 996.
Problem: Solve the definite integral ∫(sin(1) + x^2/sin(1)) dx from a = 2 to b = 4.
Solution:
To solve this integral, we can use the substitution method. Let u = sin(1), then du = cos(1) dx.
Substituting these values into the integral, we get:
∫(sin(1) + x^2/sin(1)) dx = ∫(1 + x^2/u) (1/u) du
= ∫(1/u + x^2/u^2) du
= ln|u| + (1/u^2)(x^3/3) + C
To find the definite integral from a = 2 to b = 4, we substitute the values into the antiderivative:
∫[2,4] (sin(1) + x^2/sin(1)) dx = [ln|sin(1)| + (1/sin(1)^2)(4^3/3)] - [ln|sin(1)| + (1/sin(1)^2)(2^3/3)]
= ln|sin(1)| + (1/sin(1)^2)(64/3) - ln|sin(1)| - (1/sin(1)^2)(8/3)
= (1/sin(1)^2)(64/3 - 8/3)
= (1/sin(1)^2)(56/3)
Therefore, the value of the definite integral is (56/3)(1/sin(1)^2).