Evaluate the following integrals. Show all the steps in your calculation.
a. �ç_0^2(3x^3 - x^2 + 2)dx
b. �ç_0^1(e^2x - x^2)dx
c. �ç1/(x+3)dx
Evaluate the following integrals. Show all the steps in your calculation.
a. �ç_0^2(3x^3 - x^2 + 2)dx
b. �ç_0^1(e^2x - x^2)dx
c. �ç1/(x+3)dx
a. This just a simple power rule:
∫x^n dx = 1/(n+1) x^(n+1)
b. same here, also
∫e^u du = e^u
Let u=2x so du = 2 dx
c. Recall that ∫ 1/u du = ln(u)
Let u = x+3
Don't forget the +C on indefinite integrals
a. To evaluate the integral �ç_0^2(3x^3 - x^2 + 2)dx, we can use the power rule for integration, which states that integrating a term of the form x^n gives us (1/(n+1)) * x^(n+1). Let's evaluate the integral step by step:
∫[0, 2] (3x^3 - x^2 + 2)dx
= (3/4) * x^4 - (1/3) * x^3 + 2x + C (applying the power rule)
= (3/4) * 2^4 - (1/3) * 2^3 + 2 * 2 - (3/4) * 0^4 + (1/3) * 0^3 - 2 * 0 + C
= (48/4) - (8/3) + 4 + C
= 12 - (24/3) + 4 + C
= 12 - 8 + 4 + C
= 8 + C
Therefore, the value of the integral �ç_0^2(3x^3 - x^2 + 2)dx is equal to 8 + C, where C is the constant of integration.
b. To evaluate the integral �ç_0^1(e^2x - x^2)dx, we can integrate each term separately using the power rule and the rule for the integral of e^x:
∫[0, 1] (e^2x - x^2)dx
= (1/2) * e^2x - (1/3) * x^3 + C (applying the power rule and the rule for the integral of e^x)
= (1/2) * e^2 - (1/3) * 1^3 + C
= (1/2) * e^2 - (1/3) + C
Therefore, the value of the integral �ç_0^1(e^2x - x^2)dx is equal to (1/2) * e^2 - (1/3) + C, where C is the constant of integration.
c. To evaluate the integral �ç1/(x+3)dx, we can use the logarithmic rule for integration, which states that the integral of 1/x is equal to ln|x|. Let's evaluate the integral step by step:
∫[a, b] 1/(x+3)dx
= ln|x+3| + C (applying the logarithmic rule)
Therefore, the value of the integral �ç1/(x+3)dx is equal to ln|x+3| + C, where C is the constant of integration.