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MAC1105, College Algebra!

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MAC1105, College Algebra!

Post  JFV on Tue Nov 25, 2008 9:24 pm

Write the following expression as a sum and/or difference of logarithms. Express powers as factors.

ln(xsqrt(1+x^(2))) x>0

Thank you very much!!!


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Re: MAC1105, College Algebra!

Post  bfrsoccer on Tue Nov 25, 2008 10:20 pm

ln(x) + 1/2 * ln(1+(x^2))

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Re: MAC1105, College Algebra!

Post  orten999 on Thu Apr 22, 2010 12:57 am

You can solve this question with this method.

The formula for integration by parts is derived by re-arranging the product rule for differentiation.

Let u=f(x) and v=g(x). Then du=f'(x)dx and dv=g'(x)dx, so by the substitution rule, the formula for integration by parts becomes I[,,u,v]=uv-I[,,v,u].

Break the integrand into two separate parts, u and dv.
u=ln(x), dv=(1)/(x^((1)/(2))dx)

Find du by differentiating u, and find v by integrating dv.
u=ln(x), dv=(1)/(x^((1)/(2))dx)_du=(1)/(x) dx, v=2x^((1)/(2))

Replace the values of u, v, and du in the formula for integration by parts.

Solve the second half of the integration by parts formula by finding I[,,(2)/(x^((1)/(2))),x].

Rewrite the expression to make the exponent of x negative and move it to the numerator.

To find the integral of 2x^(-(1)/(2)), find the anti-derivative. The formula for the anti-derivative of a basic monomial is I[x^(n)=(x^(n+1))/((n+1)),x].

The indefinite integral also has some unknown constant. This can be proven by completing the opposite operation (derivative) in which C would go to 0. The value of C can be found in cases when an initial condition of the function is given.

Replace the solved integral back in the formula for integration by parts.

Simplify the result.

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