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Mathematics - Calculus

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1. Write arctan ...
2. Evaluate the integrals ...
3.Use integration by parts twice to find ...

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The expert evaluates integration by parts twice to find the solution functions. Neat, step-by-step solution is provided.

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Integration

u = arctan x and dv = 1 dx
 udv = uv -  v du
 arctan x dx = arctan x * x -  [x/(1 + x^2)]dx
= x * arctan x -  (1/2) dt/t, where t = 1 + x^2
= x * arctan x - (1/2) ln t + C
= x * arctan x - (1/2) ln(1 + x^2) + C

(2) u = t and dv = sin t dt
 udv = uv -  v du
 t sin t dt = t(- cos t) -  (- cos t)dt = -t cos t +  cos t dt = -t cos t + sin t + C
(4) u = t and dv = e^(5t) dt
 udv = uv -  v du
 t e^(5t) dt = t[e^(5t) /5] - [{e^(5t) /5} dt = (1/5) t e^(5t) - (1/5)  e^(5t) dt
= (1/5) t e^(5t) - (1/25) e^(5t) + C
(18) u = z and dv = e^-z dz
 udv = uv -  v du
 z e^-z dz = z(-e^-z) - [-e^-z] dz = -z e^-z +  e^-z dz = -z e^-z - e^-z + C

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