L3 Integration
Question Bank
Find $\displaystyle\int 2\sin 5x\cos 3x\, dx$.
Find $\displaystyle\int \sin^2 3x\, dx$.
Find $\displaystyle\int 4\tan^2 2x\, dx$.
Find the exact value of
$ \int_0^{\pi/6} 2\cos 5x\cos x\, dx $
You must use calculus and show the results of any integration needed to solve the problem.
Find the exact value of
$ \int_0^{\pi/4} 6\sin 3x\sin x\, dx $
You must use calculus and show the results of any integration needed to solve the problem.
The graph shows the curve $y = 6\cos^2 2x$.
Find the exact area of the shaded region, under the curve from $x = 0$ to $x = \frac{\pi}{6}$.
You must use calculus and show the results of any integration needed to solve the problem.
The graph shows the curve $y = 4\sin 6x\cos 2x$.
Find the area of the shaded region, between the curve and the $x$-axis from $x = 0$ to $x = \frac{\pi}{6}$.
You must use calculus and show the results of any integration needed to solve the problem.
Find the exact value of
$ \int_0^{\pi/12} \left(\cos^4 x - \sin^4 x\right) dx $
You must use calculus and show the results of any integration needed to solve the problem.
Find $\displaystyle\int 8\sin^2 x\cos^2 x\, dx$.
Find the exact value of
$ \int_0^{\pi/2} \frac{\cos^2 x}{1 + \sin x}\, dx $
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = 6\cos 4x\cos 2x $
given that $y = 2$ when $x = \frac{\pi}{4}$.
Hence find the exact value of $y$ when $x = \frac{\pi}{12}$.
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = 3\tan^2 x $
given that $y = 1$ when $x = \frac{\pi}{4}$.
Hence find the exact value of $y$ when $x = \frac{\pi}{3}$.
You must use calculus and show the results of any integration needed to solve the problem.
The graph shows the curve $y = 4\sin x\cos 2x$, for $x$ from $0$ to $\frac{\pi}{2}$.
Find the total area of the shaded regions, between the curve and the $x$-axis.
Give your answer in exact form.
You must use calculus and show the results of any integration needed to solve the problem.
The graph shows the curve $y = 2\sin x\cos 3x$ for $0 \le x \le \frac{\pi}{2}.$ The curve meets the $x$-axis at the origin, at the point P and at $x = \frac{\pi}{2}.$
Find the total area of the two shaded regions between the curve and the $x$-axis. Give your answer in exact form.
You must use calculus and show the results of any integration needed to solve the problem.
The graph shows the curves $y = 2\sin^2 x$ and $y = \sin 2x$, for $x$ from $0$ to $\pi$.
Find the total area of the shaded regions enclosed between the two curves.
Give your answer in exact form.
You must use calculus and show the results of any integration needed to solve the problem.
The graph shows the curve $y = \sin^3 x$, for $x$ from $0$ to $\pi$.
The shaded region, between the curve, the $x$-axis and the line $x = k$, where $0 < k \le \pi$, has area $\frac{5}{24}$.
Find the exact value of $k$.
You must use calculus and show the results of any integration needed to solve the problem.