Why can \int\frac1x dx not use the power rule?

The rule would need \frac{x^0}{0}; the answer is \ln|x|+c instead.

How do you find c for a curve with a given gradient function?

Integrate, then substitute the coordinates of a point on the curve.

Velocity changes sign during the motion. How do you find distance?

Integrate in separate stages up to the turning point, then add the sizes of the displacements.

Integrating exponentials and trigonometric functionsAQA A-Level Maths: Flashcards

Card 1 of 120 of 12 known

Question

$\int e^{kx}dx$

Tap or press Space to reveal

Tap card or press Space to flip

See all 12 cards

$\int e^{kx}dx$: $\frac1ke^{kx}+c$
$\int\frac1x dx$: $\ln|x|+c$
$\int\sin kx\,dx$: $-\frac1k\cos kx+c$
$\int\cos kx\,dx$: $\frac1k\sin kx+c$
$\int 6e^{3x}dx$: $2e^{3x}+c$
$\int5\cos\frac x2\,dx$: $10\sin\frac x2+c$
Why can $\int\frac1x dx$ not use the power rule?: The rule would need $\frac{x^0}{0}$ ; the answer is $\ln|x|+c$ instead.
What unit must angles be in for $\int\cos kx\,dx=\frac1k\sin kx$ ?: Radians.
Simplify $e^{3\ln2}$ .: $2^3=8$ .
$\int_\pi^{2\pi}\frac3x dx$ in simplest form?: $3\ln2$
How do you find $c$ for a curve with a given gradient function?: Integrate, then substitute the coordinates of a point on the curve.
Velocity changes sign during the motion. How do you find distance?: Integrate in separate stages up to the turning point, then add the sizes of the displacements.

Exam questions on Integrating exponentials and trigonometric functions

A curve has gradient function $\frac{dy}{dx}=6e^{3x}$ and passes through the point $(0,5)$ .
Find the exact value of $\int_0^{\ln 2}6e^{3x}\,dx$ .2 marks
A particle moves on a straight line. At time $t$ seconds its velocity is $v=4\cos 2t$ m s $^{-1}$ , where $2t$ is in radians. The particle starts at the origin, so its displacement is $s=0$ when $t=0$ .
Find the total distance travelled by the particle between $t=0$ and $t=\frac{\pi}{2}$ .2 marks
Let $f(x)=\frac{3}{x}-4e^{-2x}+5\cos\frac{x}{2}$ and $h(x)=\frac{3}{x}+5\cos\frac{x}{2}$ , for $x>0$ , where angles are in radians.
Find $\int f(x)\,dx$ .3 marks

See the full worksheet

Written by the Exaim team, led by Shaun Daswani (Head of Upper Secondary, Improve ME Institute; MSc Financial Mathematics, Imperial College London; BSc, UCL) and Jason Daswani (operational lead, Improve ME Institute; LSE).