1. Which method naturally integrates x e^x?
- Integration by parts
- Add the two integrals
- Differentiate the denominator
Check the answer and reasoning
Integration by parts
A polynomial times an exponential is simplified by integration by parts.
MATH / A2 level
Advanced calculus and differential equations
Product, quotient and chain rules differentiate combined functions. Integration by substitution and by parts reverse different patterns of differentiation.
Learning objectives
- Explain advanced calculus and differential equations using a concrete example.
- Apply the worked method and justify each step.
- Recognise the limitation described in the final check.
The big idea
Product, quotient and chain rules differentiate combined functions. Integration by substitution and by parts reverse different patterns of differentiation.
A separable differential equation relates a function to its rate of change. Integrate with constants and apply an initial condition, preserving any equilibrium solutions lost by division.
Equations and relationships
dy/dx=ky gives y=Ce^(kx)
Exponential solutions for constant k.
Worked example
Differentiate sin(3x), with x in radians.
Follow the solution
The chain rule multiplies the outer derivative cos(3x) by the inner derivative 3.
Investigate and explain
Predict the worked result before calculating. Change one input, record the new result, and explain which assumptions still hold.
Related interactive model. Compare its assumptions with the explanation; it is not a complete syllabus lesson.
Check your understanding
2. Differentiate sin(3x), with x in radians.
- cos(3x)
- 3 cos(3x)
- −3 cos(3x)
Check the answer and reasoning
3 cos(3x)
The chain rule multiplies the outer derivative cos(3x) by the inner derivative 3.
3. For dy/dx=2y and y(0)=3, which solution fits?
- y=2e^(3x)
- y=3+2x
- y=3e^(2x)
Check the answer and reasoning
y=3e^(2x)
Differentiate to obtain 2y and substitute x=0 to recover 3.