Exercise 1:
Integrate the function f(x) = 3x^2
Answer:
Using the power rule, the antiderivative of x^n is (x^(n+1))/(n+1). So, the antiderivative of
3x^2 is (3x^3)/3 = x^3.
The definite integral is represented by the symbol ∫b a f(x)dx, where a and b represent the
upper and lower limits of integration. So in this case, the definite integral is ∫b a (x^3)dx =
x^3 evaluated at the upper limit minus x^3 evaluated at the lower limit.
Exercise 2:
Integrate the function f(x) = 4x + 2
Answer:
Using the constant multiple rule, the antiderivative of 4x is 4x and the antiderivative of 2 is
2. So, the antiderivative of 4x + 2 is 4x + 2.
The definite integral is represented by the symbol ∫b a f(x)dx, where a and b represent the
upper and lower limits of integration. So in this case, the definite integral is ∫b a (4x + 2)dx =
4x + 2 evaluated at the upper limit minus 4x + 2 evaluated at the lower limit.
Exercise 3:
Integrate the function f(x) = (2x^2 - 3x + 4) / (x + 1)
Answer:
We can use the substitution method in this case. Let u = x + 1, then du/dx = 1. Therefore, f(x)
= (2x^2 - 3x + 4) / (x + 1) = 2(u-1)^2 - 3u + 4.
So, the definite integral is ∫b a (2(u-1)^2 - 3u + 4)du = 2(u-1)^3/3 - 3u^2/2 + 4u evaluated at
the upper limit minus 2(u-1)^3/3 - 3u^2/2 + 4u evaluated at the lower limit, where u = x + 1.
Exercise 4:
Integrate the function f(x) = e^x
Answer:
The antiderivative of e^x is e^x. So, the definite integral is represented by the symbol ∫b a
e^xdx = e^x evaluated at the upper limit minus e^x evaluated at the lower limit.
Exercise 5:
Integrate the function f(x) = sin(x)
Answer:
The antiderivative of sin(x) is -cos(x). So, the definite integral is represented by the symbol
∫b a sin(x)dx = -cos(x) evaluated at the upper limit minus -cos(x) evaluated at the lower limit.
Exercise 6:
Integrate the function f(x) = cos^2(x)
Answer:
