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Evaluate ∫sin⁴(x) dx by expressing sin⁴(x) in terms of multiple angles

First we remember that sinθ can be expressed in terms of powers of z, where z=cos(θ)+isin(θ), using the following:2isin(nθ)=zⁿ-z⁻ⁿ and 2cos(nθ)=zⁿ+z⁻ⁿ so, [2isin(θ)]⁴=[z¹-z⁻¹]⁴ 16sin(θ)=(z)⁴(-z⁻¹)⁰+4(z)³(-z⁻...
NH
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How is a nerve impulse transmitted across a cholinergic synapse?

Action potential reaches pre-synaptic knob.This stimulates voltage gated Ca2+ channels to open and Ca2+ to diffuse in.This causes vesicles containing ACh to bind to the pre-synaptic membrane and release ACh ...
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Answered by Amelie M. • Biology tutor
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Integrate the natural logarithm of x (ln x) with respect to x

In order to integrate ln x you have to use integration by parts, even though it appears there is only one term to be integrated. We get around this by instead writing it as (ln x)(1), where we treat the 1 as...
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Answered by Archie D. • Maths tutor
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a) Express 4(cosec^2(2x)) - (cosec^2(x)) in terms of sin(x) and cos (x) and hence b) show that 4(cosec^2(2x)) - (cosec^2(x)) = sec^2(x)

A) 4(cosec 2 (2x)) - (cosec 2 (x)) = 4/(sin 2 (2x)) - 1/(sin 2 (x)) = 4/[(2 sin(x) cos(x)) 2 ] - 1/(sin 2 (x)) B) 4/[(2 sin(x) cos(x)) 2 ] - 1/(sin 2 (x)) = 4/(4 sin 2 (x) cos 2 (x)) - 1/(sin 2 (x)) = 1/(sin...
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Answered by Mario R. • Maths tutor
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2 Capacitors (c1 = 500mf) and (c2=300mf), are connected in parallel to a 10v d.c supply. Calculate the total capacitance of the circuit, and hence the total energy stored in the capacitors.

Use the equation Ct = C1 + C2, 500mF + 300mF = 800mF or 800x10^-3F 2) Appropriate equation is W=1/2 V^2 C W = (0.5)*(10^2)*800^-3 = 40J
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Answered by Samuel M. • Physics tutor
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