Lesson: Understanding Definite Integration for Entrance Exams

Problem: Evaluate the definite integral:

Solution:

Step 1: Set Up the Integral

We start by directly setting up the given integral:

To solve this integral, we will find the antiderivative of the given polynomial function.

Step 2: Integrate Each Term Individually

Break the given function into individual components and integrate term by term:

Evaluate each term:

Integral of :

Integral of :

Integral of :

Step 3: Apply the Limits of Integration

Now, substitute the limits of integration (0 and 1) into each term to evaluate the definite integral:

Evaluate each expression at the upper and lower limits:

For  from 0 to 1:

For  from 0 to 1:

For  from 0 to 1:

Step 4: Combine the Results

Now combine the evaluated terms:

Simplify the expression:

Final Answer:

Explanation Summary:

To evaluate this definite integral, we broke down the polynomial function into individual terms, integrated each term, and applied the limits of integration. Properly combining all evaluated terms led us to the final result.

Tip for Entrance Exams:

Always break down a complex polynomial into manageable parts and evaluate the integral step by step. It helps to keep your work organized and reduces the chances of making algebraic errors.