Grade 12 Advance Mathematics Examination Question : Calculus
Introduction
Calculus is a very important topic for Papua New Guinea Grade 12 students, especially those preparing for the national examinations. Mastering calculus helps students understand how quantities change, which is essential in physics, engineering, economics, health sciences, and other tertiary studies. In the Grade 12 exams, integration questions often test students on basic rules such as the power rule. Students who understand these steps clearly can earn easy marks and improve their overall performance in mathematics.
\[ y = \int 2x \, dx \]
We now solve the integral step by step.
Step 1: Identify the integrand
The expression inside the integral is:
\[ 2x \]
Step 2: Recall the power rule for integration
The power rule states:
\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \]
Step 3: Factor out the constant
The constant 2 can be taken outside the integral:
\[ \int 2x \, dx = 2 \int x \, dx \]
Step 4: Integrate \( x \)
Using the power rule:
\[ \int x \, dx = \frac{x^2}{2} \]
Step 5: Multiply by the constant
\[ 2 \times \frac{x^2}{2} = x^2 \]
Step 6: Add the constant of integration
Every indefinite integral must include a constant:
\[ x^2 + C \]
Step 7: Express the result as \( y \)
\[ y = x^2 + C \]
Final Answer:
\[ \boxed{y = x^2 + C} \]
Watch detailed video calculation here
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