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Use integration by parts to find the integral of xsinx, with respect to x

The integration by parts rule looks like this: ∫ u * v' dx = u * v - ∫ ( v * u' ) dx Hence in this example, we want to make our u = x and v' = sinx So we now need to work out what u' and v are: u' = 1 which ...
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Answered by Toby S. • Maths tutor
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Differentiate y = (6x-13)^3 with respect to x

We are looking for dy/dx, and in this question, the key rule to remember is the chain rule. Therefore the first thing to do is the easy step, take the 3 down and minus 1 from it, this will give 3(6x-13) 2 . ...
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Answered by Toby S. • Maths tutor
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Lisa, Max and Punita share £240 in the ratio 3 : 4 : 8 How much more money than Lisa does Punita get?

We should put the names aside to begin with and just look at the ratio and the price. In order to find out how much the ratio 3, the ratio 4 and the ratio 8 are worth, we need to find out how much the ratio ...
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Answered by Freddie R. • Maths tutor
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By use of calculus, show that x − ln(1 + x) is positive for all positive x.

Let f be a function defined on the positive real axis with values in R, such that f(x) = x - ln(1+x). Differentiating this function, we obtain, f'(x) = (x - ln(1+x))' = x' - ln'(1+x) = 1 - 1/(1+x). Since, x ...
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Answered by Andreea I. • STEP tutor
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How many distinct real roots does the equation x^3 − 30x^2 + 108x − 104 = 0 have?

We can see that 104 = 2^3 * 13 = 2 2 26, 30 = 2 + 2 + 26, and 108 = 2 2 + 2 26 + 2*26, so the coefficients agree with the Vieta's formulas, so the roots of the equation above are 2, 2, 26. In conclusion, it ...
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Answered by Andreea I. • MAT tutor
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