Question

Find the following indefinite integrals: (a) (4+x2)1dx\int\left(4+x^2\right)^{-1} d x; (b) (8+2xx2)1/2dx\int\left(8+2 x-x^2\right)^{-1 / 2} d x for 2x42 \leq x \leq 4; (c) (1+sinθ)1dθ\int(1+\sin \theta)^{-1} d \theta; (d) (x1x)1dx\int(x \sqrt{1-x})^{-1} d x for 0<x10<x \leq 1.

Solution

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(a):(4+x2)1dx\textbf{(a)}:\int \left(4+x^{2} \right)^{-1}dx

we can write the integral in the form and then apply standard integrals by using subsection 2.2.3:\text{we can write the integral in the form and then apply standard integrals by using subsection 2.2.3}:

=14+x2dx=1x2+22dx=12tan1(x2)+C           (1x2+a2dx=1Atan1(xa)+C)\begin{align*} &= \int \dfrac{1}{4+x^{2}}dx\\\\ &=\int \dfrac{1}{x^{2}+2^{2}}dx\\\\ &=\boxed{\dfrac{1}{2}\tan^{-1}\left(\dfrac{x}{2} \right)+C}\ \ \ \ \ \ \ \ \ \ \ \color{#c34632}\left(\int \dfrac{1}{x^{2}+a^{2}}dx=\dfrac{1}{A}\tan^{-1}\left(\dfrac{x}{a} \right)+C \right) \end{align*}

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