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How would you prove the 'integration by parts' rule?

This involves thinking about a well-known formula (the product rule) in a slightly different way. Looking at the product rule, for two functions u and v, (uv)' = uv' + vu'. We can rewrite this as uv' = (uv)'...
ER
Answered by Ethan R. • STEP tutor
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What is the largest positive integer that always divides n^5-n^3 for n a natural number.

First we note we can factorise the n^3 out of the expression, giving n^3(n^2-1). Secondly, we can see that the second term is a difference of two squares, allowing is to factorise the total to n^3(n-1)(n+1)....
WV
Answered by Ward V. • STEP tutor
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Differentiate: f(x)=(ax^2 + bx + c) ln(x + (1+x^2)^(1/2)) + (dx + e) (1 + x^2)^(1/2). Hence integrate i) ln(x + (1 + x^2)^(1/2)), ii) (1 + x^2)^(1/2), iii) x ln(x + (1 + x^2)^(1/2)).

Differentiate equation: f'(x) = (2ax + b) ln(x + (1+x^2)^(1/2)) + ((a + 2d)x^2 + (b + c)x + (c+d)) (1 + x^2)^(-1/2). Select correct values for constants to get: i) x ln(x + (1+x^2)^(1/2)) - (1 + x^2)^(1/2) +...
ME
Answered by Morgan E. • STEP tutor
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Let y=arcsin(x)/sqrt(1-x^2). Show that (1-x^2) y'-xy-1=0, and prove that, for all integers n>=0, (1-x^2)y^{n+2}-(2n+3)xy^{n+1} -(n+1)^2 y^{n}=0. (Superscripts denote repeated differentiation)

This is the first part of a STEP question (STEP 3, 2013, Q1), and is an example of a recurring pattern - "Induction Differential Equation". The first part is a computation, combining the chain and ...
DH
Answered by Daniel H. • STEP tutor
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(x_(n+1), y_(n+1))=(x_n^2-y_n^2+a, 2x_ny_n +b+2). (i) Find (x1, y1) if (a, b)=(1,-1) and (x_n, y_n) is constant. (ii) Find (a, b) if (x1, y1)=(-1,1) and (x_n, y_n) has period 2.

(i) x_1=x_1^2-y_1^2+1, y_1=2x_1y_1+1 by the equations given and the equality of x_1, x_2, x_3 . Substituting and trial and error of factor theorem results in x_1(x_1-1)(4x_1^2-4x_1+5)=0 . The quadratic has n...
TD
Answered by Tutor80806 D. • STEP tutor
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