23280 questions

How to integrate lnx by parts?

Integration by parts formula: ∫ u*dv/dx = uv - ∫ du/dx*v dx

To solve this problem we need to use a trick by thinking of lnx as lnx*1
So we can choose: u=lnx, dv/dx=1
The next step is to find du/dx and v.
du/dx=1/x                                          As we have differentiated each side with respect to x
v=x                                                         By integrating each side with respect to x
Now we have all the required parts to use the integration by parts formula.
∫ lnx = lnx*x – ∫ 1/x*x dx
                       = xlnx – ∫ 1 dx
                       = xlnx – x + c
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Ryan J.

Answered by Ryan, Maths A Level tutor with MyTutor


How do you find the inverse of a function?

So you are asked to find the inverse of a function f(x).
The inverse function is denoted by f -1(x).
To help with this we can use the identity f(f -1(x))=x.
Now, we need to define y=f -1(x).
x=f(f -1(x))=f(y)=2y+1                    As f(y) is similar to f(x) but with the variable change of x to y
Hence, we need to solve:           
x-1=2y                                                  Minus 1 from each side of the equation
½(x-1)=y=f -1(x)                                                As we defined f -1(x)=y
Therefore, we have found the inverse function: f -1(x) = ½(x-1)

We can continue further and find the domain and range of an inverse function using the identities:
Domain f(x) = Range f -1(x)
Range f(x) = Domain f -1(x)
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Ryan J.

Answered by Ryan, Maths A Level tutor with MyTutor


What are the differences between arithmetic and geometric sequences?

An arithmetic sequence has a constant difference between each term.
For example: 2,4,6,8,10,12,…
We can see clearly that all the terms differ by +2.
We call this the common difference, d. A geometric sequence has a constant ratio (multiplier) between each term.
An example is: 2,4,8,16,32,…
So to find the next term in the sequence we would multiply the previous term by 2.
This is called the common ratio, r. These sequences are closely related as they both have the same first term, but I hope you can see how different they become if they have a common difference or a common ratio.
We can create a decreasing arithmetic sequence by choosing a negative common difference.
Similarly, a decreasing geometric sequence would have a common ratio of less than 1. 
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Ryan J.

Answered by Ryan, Maths GCSE tutor with MyTutor


What is the quotient rule and how is it applied?

The quotient rule is a formula used to differentiate fractions that contain terms of x in both the numerator and the denominator (we usually refer to this as f(x)/g(x)). As it isn't possible to differentiate this by normal methods straight away, we use the quotient rule to allow us to do so. For A-level maths, you only need to know how to use the quotient rule, deriving how it is formed is not necessary. The quotient rule takes the following form: dy/dx = ((g(x)f'(x) - f(x)g'(x))/g(x)^2 Where f'(x), g'(x) are dy/dx of f(x), g(x) respectively. Therefore differentiating the numerator and the denominator seperately, and then plugging these back into the formula, will yield the result of differentiating a fraction of the form f(x)/g(x).
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Pete H.

Answered by Pete, Maths A Level tutor with MyTutor


How can I analyse a quote using close reading?

It is important to use quotations as evidence in your essay, but you also need to use them in a way that effectively backs up your argument, rather than simply inserting them after a statement. A good way to add to your explanation is through close reading, in which you examine and analyse the quote in detail. Here are a few points to consider when close reading. First impression: To begin with, think about your immediate response to the quotation you’ve chosen – what are the first things that stand out about it? What is its overall effect? Why is it significant? Language: Look at the language used, focusing on individual words. What kind of imagery do they create? Do any words have particular connotations? Do any allow for contrasting interpretations – and if so, which reading do you think is correct? Make a note of any devices that are used: e.g. metaphor or simile, alliteration, rhyme, repetition etc. What effect are these used to create? Structure: Does the word order affect your response, and how? You can also look at the length of the sentence and the way punctuation is used. For example, a very short sentence might be used for dramatic effect. Or, in a long sentence, a crucial piece of information might be withheld until the end of a sentence, creating suspense as the reader waits to find it out. Link to the question: Keep in mind that this should all be relevant to the essay you are writing. You should avoid just listing what’s there, and instead talk about the way it affects the rest of the text, how it relates to the context, and how it supports your argument. Looking closely at a small quote is a useful way to illustrate your points, but remember to answer the question, and link your close reading to your response.
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Melissa T.

Answered by Melissa, who tutored English A Level with MyTutor


What is Gibbs free energy? How is it useful?

The Gibbs free energy of a process is a measure of how readily that process will proceed on its own - a negative value means that the reaction is spontaneous, a positive value means that it is not, and a value of or very close to zero means that it's an equilibrium process.
  It's equation is: G = H - TS
Where G is the Gibbs energy, H is the enthalpy change, T is the temperature that the process is at, and S is the entropy change.
Temperature here is in Kelvin, and a classic trick is to give you the temperature in celsius so you have to remember to account for it by adding 273.
Another easy pitfall is that since the entropy S is so much smaller than the enthalpy H, it will be in J K-1 Mol-1 - and H will be in kJ K-1 Mol-1​. So you need to put both values in the same units. Always remember to check that your units 'add up' correctly - this is just as important as your numbers. Typically a question will ask you to find the enthalpy change, or entropy change of a reaction, then give you the value you don't have and ask you to comment on how likely the reaction is to occur through calculating this Gibbs energy. These questions are, when it comes down to it, just a bit of arithmetic. 
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Murdo M.

Answered by Murdo, who has applied to tutor Chemistry A Level with MyTutor


What is the pluperfect tense and how is it formed?

Subject: French/A Level Q: How do I form the pluperfect and when do I use it?  A:  In French, the pluperfect tense is used to describe a completed event that has taken place before another event in the past. This is equivalent to the English 'had', for example:  Before he came to my house, he had already eaten.  You can remember the function of the pluperfect tense by thinking of its French title, plus-que-parfait, which literally translates as 'more than perfect.' Therefore, the title refers to the fact that this tense describes events that are 'more completed' as they occur before the perfect tense.  In order to form the pluperfect tense, you will need to know: 
1. The French Subject Pronouns: Je, Tu, Il/Elle/On, Nous, Vous, Ils/Elles
2. The auxiliary verbs avoir and être in the imperfect tense: 
Avoir: j'avais, tu avais, il/elle/on avait, nous avions, vous aviez, ils/elles avaient
Être: j'étais, tu étais, il/elle/on était, nous étions, vous étiez, ils/elles étaient
3. The past participle of the verb in question (e.g. manger becomes mangé) and which auxiliary (avoir or être) it requires. 
4. The fact that verbs taking être and preceeding direct objects must agree with the subject in gender and quantity. For example: Elle était sortie avant qu'il n'est arrivé and Les maisons que j'avais achetées Therefore, the formation of the pluperfect is very similar to that of the perfect. To decide which auxiliary a verb requires, you will need to distinguish between the verbs that take être, which can be remembered by the DR MRS VANDERTRAMP acrostic, and all other verbs which take aller. 
Devenir - Devenu
Revenir - Revenu 

Monter - Monté
Rester - Resté
Sortir - Sorti

Venir - Venu
Aller - Allé
Naitre - Né
Descendre - Descendu
Entrer - Entré
Rentrer - Rentré
Tomber - Tombé
Retourner - Retourné
Arriver - Arrivé
Mourir - Mort
Partir - Parti Examples of the pluperfect: 
J'étais née avant...
Nous avions quittés la maison avant... 
Elles étaient montées...
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Emily W.

Answered by Emily, French A Level tutor with MyTutor


What makes an A* EPQ?

Getting an A* on your EPQ begins with picking a creative, individual and exciting question that is both specific and sufficiently broad to provide a detailed response. Above all, the most successful EPQs are those that genuinely interest the participant.  In order to create such a question, think ahead. Create mindmaps and document plenty of research. From here, you need to refine your ideas. Successful EPQs demonstrate progression and development from the initial collection of vague ideas to the polished final product.  Additionally, you must remember that the process of creating the project is as important, if not more important than the final product. Therefore, ensure that your log book is always up to date and completed as thouroughly as possible. This will entail continuous evaluation  Furthermore, use a variety of research methods, as carefully considered primary and secondary research will always impress. My EPQ involved comparing approaches to the translations of Mallarmé's poetry, therefore I used letters and E-Mail correspondances with translators as an appendix to my project.  Finally, don't overlook the presentation. For many candidates, the presentation constitutes both the final hurdle and the most daunting element. Therefore, be organised, practise presenting, create cue cards and an eyecatching powerpoint. These aspects are not necessary components, but are certainly helpful to a confident delivery. To prepare really thoroughly, you could consider questions you might be asked in the Q&A which follows the presentation and practise answering spontaneously. 
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Emily W.

Answered by Emily, Extended Project Qualification A Level tutor with MyTutor

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