Integrating Trigonometric Functions by Recognition
Example 1
Find the derivative of \( \sin (2x-5) \) and use this result to deduce \( \displaystyle \int 10 \cos (2x-5) dx \).
\( \begin{align} \displaystyle \frac{d}{dx} \sin (2x-5) &= \cos (2x-5) \times \frac{d}{dx} (2x-5) &\color{green}{\text{Chain Rule}} \\ &= \cos (2x-5) \times 2 \\ &= 2 \cos (2x-5) \\ \sin (2x-5) &= \int 2 \cos (2x-5) dx \\ \int 10 \cos (2x-5) dx &= 5 \int 2 \cos (2x-5) dx \\ \require{AMSsymbols} \therefore \int 10 \cos (2x-5) dx &= 5 \sin (2x-5) + C \end{align} \)
Example 2
Find the derivative of \( x \cos x \) and use this result to find \( \displaystyle \int x \sin x dx \).
\( \begin{align} \displaystyle \frac{d}{dx} x \cos x &= x \times \frac{d}{dx} \cos x + \frac{d}{dx} x \times \cos x &\color{green}{\text{Product Rule}} \\ &= x \times (-\sin x) + 1 \times \cos x \\ &= -x \sin x + \cos x \\ x \cos x &= \int (-x \sin x + \cos x) dx \\ x \cos x &=-\int x \sin x dx + \int \cos x dx \\ \int x \sin x dx &= \int \cos x dx-x \cos x \\ \require{AMSsymbols} \therefore \int x \sin x dx &= \sin x-x \cos x + C &\color{green}{\int \cos x dx = \sin x + C} \end{align} \)
Algebra Algebraic Fractions Arc Binomial Expansion Capacity Common Difference Common Ratio Differentiation Double-Angle Formula Equation Exponent Exponential Function Factorise Functions Geometric Sequence Geometric Series Index Laws Inequality Integration Kinematics Length Conversion Logarithm Logarithmic Functions Mass Conversion Mathematical Induction Measurement Perfect Square Perimeter Prime Factorisation Probability Product Rule Proof Pythagoras Theorem Quadratic Quadratic Factorise Ratio Rational Functions Sequence Sketching Graphs Surds Time Transformation Trigonometric Functions Trigonometric Properties Volume
Responses