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How do you integrate ln(x) with respect to x?

Rewrite ln(x) as 1 ln(x) then integrate by parts. The formula for integration by parts is ∫ uv' = uv - ∫ vu', here use u = ln(x) and v' = 1. By differentiating u we get u' = 1/x, and by integrating v' we get...
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Answered by Anthony G. • Maths tutor
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What is 2/3 + 1/4?

When adding or subtracting fractions, we must always start by putting the fractions over the same denominator , that is the number on the bottom half of the fraction. To do this, we need to find a number whi...
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Answered by Catherine C. • Maths tutor
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Differentiate the function f(x) = sin(x)/(x^2 +1) , giving your answer in the form of a single fraction. Is x=0 a stationary point of this curve?

The key concepts to apply in this question will be the product and chain rules, namely: if f(x)=g(x)h(h), then f'(x)=g(x)h'(x) + g'(x)h(x), and if h(x)=u(v(x)), then h'(x)=u'(v(x))v'(x). Equivalently, you ma...
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Answered by Bromlyn C. • Maths tutor
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Show that: [sin(2a)] / [1+cos(2a)] = tan(a)

We start by expanding out the double trigonometric terms (sin(2a)) using the double angle formula, giving us: [2sin(a)cos(a)] / [1+cos^2(a) - sin^2(a)]. Next we spot that on the denominator (bottom half of f...
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Answered by George H. • Maths tutor
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What does it mean when I get a negative value when I do a definite integral?

The definite integral gives the value of the area bounded between the curve at the given limits, and the x-axis (if we are integrating with respect to x!) If this value is negative, it means that the area un...
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Answered by Callum L. • Maths tutor
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