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 \( \displaystyle \int x \sin x dx \).

\( \begin{align} \displaystyle \frac{d}{dx} x \cos x &= x \times \frac{d}{dx} \cos x + \frac{d}{dx} x \times \cos x &\color{green}{\text{Product Rule}} \\ &= x \times (- \sin x) + 1 \times \cos x \\ &= -x \sin x + \cos x \\ x \cos x &= \int (-x \sin x + \cos x) dx \\ x \cos x &= – \int x \sin x dx + \int \cos x dx \\ \int x \sin x dx &= \int \cos x dx – x \cos x \\ \require{AMSsymbols} \therefore \int x \sin x dx &= \sin x – x \cos x + C &\color{green}{\int \cos x dx = \sin x + C} \end{align} \)

Absolute Value Algebra Arithmetic Mean Arithmetic Sequence Binomial Expansion Binomial Theorem Chain Rule Circle Geometry Combinations Common Difference Common Ratio Compound Angle Formula Compound Interest Cyclic Quadrilateral Differentiation Discriminant Double-Angle Formula Equation Exponent Exponential Function Factorials Functions Geometric Mean Geometric Sequence Geometric Series Inequality Integration Kinematics Logarithm Logarithmic Functions Mathematical Induction Probability Product Rule Proof Quadratic Quotient Rule Rational Functions Sequence Sketching Graphs Surds Transformation Trigonometric Functions Trigonometric Properties VCE Mathematics Volume


Your email address will not be published. Required fields are marked *