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Integrate x * sin(x) with respect to x by using integration by parts

The general formula for integration by parts to integrate something of the form u * v' is: u * v - (integral)[ (u' * v) dx ] . Thus we first need to write x * sin(x) in the form u * v' . Lets pick u = x and ...
JW
Answered by Jamie W. • Maths tutor
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write 2(sin^2(x)- cos^2(x)) + 6 sin(x) cos(x) in terms of cos(2x) and sin(2x)

We use the following double angle formulae cos(2x) = cos^2(x) - sin^2 (x) and sin(2x) = 2sin(x)cos(x) to see that 2(sin^2(x)- cos^2(x)) + 6 sin(x) cos(x) = -2-(sin^2(x)+ cos^2(x)) + 3*2 sin(x) cos(x) = -2cos...
NR
Answered by Nicola R. • Maths tutor
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Given that 3^(-3/2) = a* 3^(1/2), find the exact value of a.

3^(3/2) = a * 3^(1/2) 3^(1/2) * 3^(-2) = 3^(3/2)Therefore a must be equal to 3^(-2)a = 1/9
OW
Answered by Oliver W. • Maths tutor
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A curve with equation y = f(x) passes through the point (4,25). Given that f'(x) = (3/8)*x^2 - 10x^(-1/2) + 1, find f(x).

f'(x) = (3/8) x^2 - 10x^(-1/2) + 1Each term must be integrated (increase the power by 1 and divide by the new power), remembering to include + c.f(x) = (3/8) (x/3)^3 - 10*(2x)^(1/2) + x + cf(x) = (1/8) x^3 -...
OW
Answered by Oliver W. • Maths tutor
9056 Views

Use the substitution u=1+e^x to find the Integral of e^(3x) / (1 + e^x)

e x =u-1 so e 3x =(u-1) 3 and du/dx = e x so rearranging gives dx=e -x du Substituting all that information in the integral we get Integral ( (u-1) 3 / (u(u-1)) du ) which simplifies to Integral (u -2 +1/u)....
IP
Answered by Ismet P. • Maths tutor
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