# Mathematics - Calculus

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1. Write arctan ...

2. Evaluate the integrals ...

3.Use integration by parts twice to find ...

[See the attached questions file.]

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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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