← All labsMathematics · Mathematics 12 · Exercise 3.1 Q3(xi)
An Animated, Step-by-Step Solution
Mathematics 12, Exercise 3.1 Q3(xi): ∫(3x + 3ˣ + x³) dx, solved in six animated steps. Every term keeps its colour, each rule is explained beside the working, and the answer is checked by differentiating.
WHAT YOU WILL DISCOVER
- Split an integral with the sum rule
- Apply the power and exponential rules
- Check an answer by differentiating
How to use this lab (7 steps)
- A real textbook question. Evaluate ∫(3x + 3ˣ + x³) dx. The storyboard begins by reading the expression and naming its three terms.
- Inspect a term. Every term keeps its colour. Tapping 3ˣ explains why it needs the exponential rule, not the power rule.
- Watch the transformation. Step 2: the sum rule gives each term its own integral sign and dx. The animation moves one term at a time.
- Move the constant. Step 3: the 3 in 3x moves outside the integral. The 3 in 3ˣ is a base, so it stays put.
- Apply the rules. Step 4: the powers gain one and 3ˣ is divided by ln 3. Why each rule works is explained beside the working.
- Simplify. Step 5 tidies the coefficients into the final answer.
- Check by differentiating. Step 6 differentiates the answer to get back the integrand. The solution is checked, not just stated.
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