Trigonometric Integrals
Integrals of powers of sin and cos.
Key idea
Integrals of powers of sin and cos.
In depth
Use identities to integrate powers of sin and cos: save one factor for du and convert the rest with sin² + cos² = 1; use power-reduction for even powers.
Worked example
∫ sin³x dx = ∫(1 − cos²x) sin x dx = −cos x + cos³x/3 + C.
Formula
Real-world uses
- Computing average power in AC electrical circuits and the energy of vibrating systems
Common mistake
Missing the identity step and getting stuck.
How to learn this
- Read the key idea and say it in your own words.
- Work through the worked example, then try a similar problem.
- Use the interactive lesson’s visual and mini-quiz to check the pattern.
Frequently asked questions
What is Trigonometric Integrals?
Integrals of powers of sin and cos. It is a topic at the expert level in Calculus.
What is the formula for Trigonometric Integrals?
The key formula is ∫sin x dx = −cos x + C.
Where is Trigonometric Integrals used in real life?
Real-world examples: Computing average power in AC electrical circuits and the energy of vibrating systems.
What should I learn before Trigonometric Integrals?
Make sure you are comfortable with Integration by Parts, the previous topic in Calculus.
What is a common mistake with Trigonometric Integrals?
Missing the identity step and getting stuck.
How can I practise Trigonometric Integrals?
Open the interactive lesson, play with the visual, then answer the three-question mini-quiz. Getting two right marks the topic complete.
What comes after Trigonometric Integrals?
Move on to Trigonometric Substitution next.
Related expert topics in Calculus
- Derivatives of Inverse Trig – Derivatives of arcsin, arccos, arctan.
- Implicit Differentiation – Differentiate both sides of F(x,y)=0 treating y as a function of x.
- Logarithmic Differentiation – Take logs before differentiating products and powers.
- Related Rates – Link rates of change of connected quantities.
- Linear Approximation – Approximate f near a using its tangent line.
- Differentials – Use dy = f′(x)dx for small changes.