Integrate the following with respect to $x$: (i) $\left(1+x^{2}\right)^{-1}$ (ii) $\left(1-x^{2}\right)^{-1/2}$

Integrate the following with respect to x: (i) (1+x²)⁻¹ (ii) (1-x²)-1/2

Exercise 11.1 · Q4

x

(i)

\left(1+x^{2}\right)^{-1}

(ii)

\left(1-x^{2}\right)^{-1/2}

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Rewriting each negative exponent as a fraction reveals a direct match to the two inverse-trigonometric rows of the standard-integrals table.

Step 1. Part (i). $(1+x^2)^{-1}=\dfrac{1}{1+x^2}$ , and $\displaystyle\int \frac{dx}{1+x^2}=\tan^{-1}x+c$ .

\int (1+x^2)^{-1}\,dx=\tan^{-1}x+c.

Step 2. Part (ii). $(1-x^2)^{-1/2}=\dfrac{1}{\sqrt{1-x^2}}$ , and $\displaystyle\int \frac{dx}{\sqrt{1-x^2}}=\sin^{-1}x+c$ .

$\int (1-x^2)^{-1/2}\,dx=\sin^{-1}x+c.$ …