13.03 Antidiffferentiation and trigonometric functions
Find the primitive function of the following:
f'(x)=7 \sin x
f'(x)=- 7 \cos x
f'(x)=6 \sin x - \cos x
f'(x)=5 \sin \left(\dfrac{x}{3}\right) - 2 \cos \left(\dfrac{x}{4}\right)
Find the following indefinite integrals:
\int \sin 7 x \ dx
\int \cos 6 x \ dx
\int \sin \left(\dfrac{x}{3}\right)dx
\int 9 \sin 3 x \ dx
\int - 5 \cos \left(\dfrac{x}{4}\right)dx
\int \cos \left(x + \dfrac{\pi}{6}\right)dx
\int \sin \left(x - \dfrac{\pi}{6}\right)dx
\int \sin \left( 5 x - \dfrac{\pi}{4}\right)dx
\int \cos \left(\dfrac{x}{8} + \dfrac{\pi}{3}\right)dx
\int 8 \cos \left( 4 x - \dfrac{\pi}{3}\right) dx
\int - 8 \sin \left( 2 x + \dfrac{\pi}{6}\right)dx
\int 2 \sin \left(\dfrac{x}{7} - \dfrac{\pi}{4}\right)dx
\int - 4 \cos \left(\dfrac{x}{7} + \dfrac{\pi}{4}\right)dx
\int \left(x^{\frac{1}{4}} + \sin 3 x\right) dx
\int \left(\cos x - \sin 3 x\right) dx
\int \left(\sin \left(\dfrac{x}{3}\right) + \cos \left(\dfrac{x}{3}\right)\right) dx
\int 7 \left(\sin 7 x + 9 \cos 3 x\right)dx
Consider the function y=x \sin x.
Find \dfrac{d}{dx} \left( x \sin x\right).
Hence find \int 4 x \cos x \,dx.
Given the gradient function and a point on the curve, find y in terms of x:
\dfrac{d y}{d x} = - 3 \sin \left(\dfrac{x}{2}\right) and \left(2 \pi, 5 \right).
\dfrac{d y}{d x} = \sin 2 x + \cos 3 x and \left(\dfrac{\pi}{2}, 4 \right).
Consider f \rq \left( x \right) = k \cos 3 x for some constant k, with f \rq \left( 0 \right) = 2 and f \left( \dfrac{\pi}{6} \right) = 6.
Find the value of k.
Hence find f \left( x \right).
Consider f \rq \left( x \right) = k \sin \left(\dfrac{x}{4}\right) for some constant k, with f \left( 2 \pi \right) = 1 and f \left( 0 \right) = - 12.
Determine the value of k.
Hence find f \left( x \right).
Consider the gradient function \dfrac{d y}{d x} = 8 \cos 2 x.
Find \int 8 \cos 2x \ dx.
Hence find y, if y = 3 when x = \dfrac{\pi}{4}.