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Integrate the function f(x)=lnx

This question must be attempted by integration by parts since it cannot be integrated outright and we can thus change the integral to 1 times lnx. We can then use the formula for integration by parts of I(in...
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Answered by Srikant S. • Maths tutor
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How do I integrate ∫ xcos^2(x) dx ?

First, recall that when integrating, squared trigonometric functions often cause issues. Therefore, use the identity: cos^2(x) = (cos(2x) + 1)/2 to remove this power, giving the integral: ∫ (xcos(2x))/2 + x/...
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Answered by Stephi S. • Maths tutor
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Chris claims that, “for any given value of x , the gradient of the curve y=2x^3 +6x^2 - 12x +3 is always greater than the gradient of the curve y=1+60x−6x^2” . Show that Chris is wrong by finding all the values of x for which his claim is not true.

(Question from AQA A level maths specimen papers) When looking at this question, you need to appreciate the steps required in order to reach the final answer before diving straight in. Firstly, the question ...
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Answered by Michael N. • Maths tutor
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Express (5sqrt(3)-6)/(2sqrt(3)+3) in the form m+nsqrt(3) where m and n are integers. [Core 1]

Step 1: multiply (5sqrt(3)-6)/(2sqrt(3)+3) by (2sqrt(3)-3)/(2sqrt(3)-3) (this is a trick called CONJUGATION as you're really just multiplying the fraction by 1. Step 2: Expand and simplify top and botton to ...
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Answered by Mahir Y. • Maths tutor
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simplify (3x^2 - x - 2) / (x^2 - 1)

factorise the top fraction: 3x^2 - x - 2 = (3x + 2)(x - 1) factorise the bottom fraction: x^2 -1 = (x + 1)(x - 1) cancel out the (x - 1) and the final result is (3x + 2) / (x + 1)
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Answered by Alice V. • Maths tutor
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