Evaluate Definite Integral of 3x^2 + 4x - 5 from 1 to 2

Published on April 21, 2026

Calculus

Learn how to solve definite integrals using the power rule and the Fundamental Theorem of Calculus. This example evaluates the integral of 3x^2 + 4x - 5 from x=1 to x=2.

Question

Answer

The problem asks us to evaluate the definite integral:

To evaluate this definite integral, we first find the antiderivative of the function

.
Using the power rule for integration, which states that

(where

), we integrate each term:

The antiderivative of

.
The antiderivative of

So, the indefinite integral is:

Now, we need to evaluate this antiderivative from the lower limit

to the upper limit

. We use the Fundamental Theorem of Calculus, which states that

, where

is the antiderivative of

Let

.
We need to calculate

First, evaluate

Next, evaluate

Finally, subtract

from

The final answer is

Theorems

• Power Rule for Integration

The power rule for integration states that for any real number

, the integral of

with respect to

is given by

, where

is the constant of integration.

• Fundamental Theorem of Calculus (Part 2)

The second part of the Fundamental Theorem of Calculus states that if

is an antiderivative of a continuous function

on the interval

, then the definite integral of

from

is given by

🚀 Try Free Math Help Now!🏠 Learn More Powerful Features