Basic Mathematics20805 marksNumerical: definite integral evaluationTrigonometric integrals
Evaluate int0fracpi2(sin 2xcos 3x + cos 2xsin 3x)dx and int0fracpi4 fracdx1 - sin x.
Evaluate $\int_0^{\frac{\pi}{2}}(\sin 2x\cos 3x + \cos 2x\sin 3x)dx$ and $\int_{0}^{\frac{\pi}{4}} \frac{dx}{1 - \sin x}$. [5]
- Integral 1: $\displaystyle\int0^{\pi/2}(\sin 2x\cos 3x + \cos 2x\sin 3x),dx$ - Integral 2: $\displaystyle\int0^{\pi/4}\frac{dx}{1-\sin x}$ --- Step 1: Apply sine addition formula $$\sin A\cos B + \cos A\sin B = \sin(A+B)$$ With $A = 2x$, $B = 3x$: $$\sin 2x\cos 3x + \cos 2x\sin 3x = \sin 5x$$ ...