L3 Integration
Question Bank
Find $\displaystyle\int 6x(x^2 + 4)^5\, dx$.
Find the exact value of
$ \int_0^2 x\sqrt{2x^2 + 1}\, dx $
You must use calculus and show the results of any integration needed to solve the problem.
Find $\displaystyle\int \frac{10x}{(x^2 + 3)^3}\, dx$.
Find $\displaystyle\int \frac{6x + 1}{2x - 1}\, dx$.
Find $\displaystyle\int \frac{x + 1}{x^2 + 2x + 7}\, dx$.
Find the exact value of
$ \int_0^{\pi/6} \sin^2 3x\cos 3x\, dx $
You must use calculus and show the results of any integration needed to solve the problem.
Find
$ \int \frac{e^{2x} - e^{-2x}}{e^{2x} + e^{-2x}}\, dx $
Find $\displaystyle\int \frac{2x^2 + x - 4}{x - 1}\, dx$.
Find the exact value of
$ \int_2^5 \frac{3x + 4}{x - 1}\, dx $
You must use calculus and show the results of any integration needed to solve the problem.
Find the exact value of each integral.
(a) $\displaystyle\int_0^1 \frac{2x}{x^2 - 4}\, dx$
(b) $\displaystyle\int_0^1 \frac{2x}{(x^2 - 4)^2}\, dx$
You must use calculus and show the results of any integration needed to solve the problem.
Solve the differential equation
$ \frac{dy}{dx} = (4x - 2)e^{x^2 - x} $
given that $y = 5$ when $x = 1$.
Hence find the exact value of $y$ when $x = 2$.
You must use calculus and show the results of any integration needed to solve the problem.
The graph shows the curve
$ y = \frac{2x - 4}{x + 1} $
which crosses the $x$-axis at $A$.
Find the exact area of the shaded region, which is bounded by the curve, the $x$-axis and the line $x = 5$.
You must use calculus and show the results of any integration needed to solve the problem.
Find the exact value of
$ \int_0^{\pi/3} \frac{\sin 3x}{2 + \cos 3x}\, dx $
You must use calculus and show the results of any integration needed to solve the problem.
Find the exact value of
$ \int_0^2 \frac{2x + 1}{x - 3}\, dx $
You must use calculus and show the results of any integration needed to solve the problem.
A short film is posted online. $t$ days after it is posted, it is being watched at a rate modelled by
$ \frac{dV}{dt} = \frac{k\ln(t + 1)}{t + 1} $
thousand views per day, where $V$ is the total number of views (in thousands) and $k$ is a constant.
In the first 2 days after it was posted, the film was viewed 30 000 times.
How many days after it was posted had it been viewed 120 000 times in total?
You must use calculus and show the results of any integration needed to solve the problem.
The graph shows part of the curve
$ y = \frac{2x - 1}{x + 1}, \quad x > -1 $
and the line $y = 3 - x$. They meet at the point $P$.
Find the exact area of the shaded region, which is bounded by the curve, the line and the $x$-axis.
You must use calculus and show the results of any integration needed to solve the problem.
The graph shows the curve
$ y = e^{\sin^2 x}\sin 2x $
for $0 \le x \le \frac{\pi}{2}$.
The shaded region, between the curve, the $x$-axis and the line $x = k$, where $0 < k \le \frac{\pi}{2}$, has area $e^{3/4} - 1$.
Find the exact value of $k$.
You must use calculus and show the results of any integration needed to solve the problem.