Find the primitive of the following:
7 \sin x
- 7 \cos x
\sin 7 x
\cos 6 x
9 \sin 3 x
\sin \left(\dfrac{x}{3}\right)
- 5 \cos \left(\dfrac{x}{4}\right)
Find the following indefinite integrals:
Find the exact value of the following definite integrals:
Consider the function f(x)=\cos ^{5}x.
Find \dfrac{d}{dx} \left(\cos ^{5}x\right).
Hence find \int_{0}^{\pi} \sin x \cos ^{4}x \, dx.
Consider the function y = \tan x on the interval 0 \leq x < \dfrac{\pi}{2}.
Rewrite the equation of the function in terms of \sin x and \cos x.
Hence find \int \tan x \,dx.
Given that \sin 3 t = 3 \sin t - 4 \sin ^{3}t, find the indefinite integral \int 3 \sin ^{3}t \ dt.
Use the identity \tan x = \dfrac{\sin x}{\cos x} to show that \int_0^\frac{\pi}{4} \tan x \, dx = \dfrac{1}{2}\ln 2.
Use the identity \cot x = \dfrac{\cos x}{\sin x} \text{ to find }\int_\frac{\pi}{6}^\frac{\pi}{2} \cot x \, dx .