Math  /  Calculus

QuestionSolve on separate sheets with signatures. No calculators. Justify answers.
1. (a) Find the integral: xarctan(x)dx\int x \arctan (x) d x (b) Evaluate: 0π/2cos(t)sin(t)dt\int_{0}^{\pi / 2} \frac{\cos (t)}{\sqrt{\sin (t)}} d t

Studdy Solution
This integral is a standard integral and its value is2.
0π/2sinucosudu=2\int_{0}^{\pi /2} \frac{\sin u}{\sqrt{\cos u}} du =2So, the solutions are(a) xarctan(x)dx=2x2arctan(x)4(+x2)+C\int x \arctan (x) dx = \frac{}{2} x^2 \arctan(x) - \frac{}{4} ( + x^2) + C(b) 0π/2cos(t)sin(t)dt=2\int_{0}^{\pi /2} \frac{\cos (t)}{\sqrt{\sin (t)}} dt =2

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