Find the integrals of the following: (i) dfrac14-x² (ii) dfrac125-4x² (iii) dfrac19x²-4

Exercise 11.10 · Q1

Q. Find the integrals of the following:

(i)

\dfrac1{4-x^2}

(ii)

\dfrac1{25-4x^2}

(iii)

\dfrac1{9x^2-4}

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Concept understanding — Integration of Rational Algebraic Functions

Several rational and irrational integrands reduce to a small table of standard forms.

Type I — pure quadratic denominators. With constant $a>0$ :

\int\frac{dx}{a^2-x^2}=\frac1{2a}\log\left|\frac{a+x}{a-x}\right|+c,\quad \int\frac{dx}{x^2-a^2}=\frac1{2a}\log\left|\frac{x-a}{x+a}\right|+c,\quad \int\frac{dx}{a^2+x^2}=\frac1a\tan^{-1}\!\frac{x}{a}+c,

\int\frac{dx}{\sqrt{a^2-x^2}}=\sin^{-1}\!\frac{x}{a}+c,\quad \int\frac{dx}{\sqrt{x^2-a^2}}=\log\left|x+\sqrt{x^2-a^2}\right|+c,\quad \int\frac{dx}{\sqrt{x^2+a^2}}=\log\left|x+\sqrt{x^2+a^2}\right|+c.

Type II — general quadratic denominators $\int\frac{dx}{ax^2+bx+c}$ and $\int\frac{dx}{\sqrt{ax^2+bx+c}}$ : make the coefficient of $x^2$ unity, complete the square to write $ax^2+bx+c$ as a sum/difference of two squares, and reduce to Type I.

Type III — linear over quadratic $\int\frac{px+q}{ax^2+bx+c}\,dx$ (and the $\sqrt{\ }$ version): write $px+q=A\frac{d}{dx}(ax^2+bx+c)+B$ , find $A,B$ by matching coefficients, then split into a log/derivative-over-function piece plus a Type-I/II piece.

Type IV — square roots of quadratics (Result 11.3), via integration by parts:

\int\sqrt{a^2-x^2}\,dx=\frac x2\sqrt{a^2-x^2}+\frac{a^2}2\sin^{-1}\!\frac xa+c,

\int\sqrt{x^2-a^2}\,dx=\frac x2\sqrt{x^2-a^2}-\frac{a^2}2\log\left|x+\sqrt{x^2-a^2}\right|+c,\quad \int\sqrt{x^2+a^2}\,dx=\frac x2\sqrt{x^2+a^2}+\frac{a^2}2\log\left|x+\sqrt{x^2+a^2}\right|+c.

Tip

Match the sign pattern first — $a^2-x^2$ leads to $\sin^{-1}$ , $a^2+x^2$ to $\tan^{-1}$ or a $\log(x+\sqrt{\ })$ , and $x^2-a^2$ to a $\log$ — before completing the square or splitting the numerator.

Match each denominator to the Type I forms $a^2-x^2$ or $x^2-a^2$ (rescaling $x$ first if the $x^2$ -coefficient isn't $1$ ) and apply the standard log formula.

(i) Let $I=\displaystyle\int\frac{dx}{4-x^2}$ .

Step 1. This is exactly Type I, $\displaystyle\int\frac{dx}{a^2-x^2}=\frac1{2a}\log\left|\frac{a+x}{a-x}\right|+c$ , with $a=2$ .

Step 2. $I=\dfrac1{2(2)}\log\left|\dfrac{2+x}{2-x}\right|+c=\dfrac1{4}\log\left|\dfrac{2+x}{2-x}\right|+c$ .

(ii) Let $I=\displaystyle\int\frac{dx}{25-4x^2}$ .

Step 1. Put $u=2x$ , so $du=2\,dx$ , and $25-4x^2=25-u^2$ : $I=\dfrac12\displaystyle\int\frac{du}{25-u^2}$ .

Step 2. Type I with $a=5$ : $I=\dfrac12\cdot\dfrac1{10}\log\left|\dfrac{5+u}{5-u}\right|+c=\dfrac1{20}\log\left|\dfrac{5+2x}{5-2x}\right|+c$ .

(iii) Let $I=\displaystyle\int\frac{dx}{9x^2-4}$ .

Step 1. Put $u=3x$ , so $du=3\,dx$ , and $9x^2-4=u^2-2^2$ : $I=\dfrac13\displaystyle\int\frac{du}{u^2-4}$ .

Step 2. Type I with $a=2$ : $I=\dfrac13\cdot\dfrac1{4}\log\left|\dfrac{u-2}{u+2}\right|+c=\dfrac1{12}\log\left|\dfrac{3x-2}{3x+2}\right|+c$ .

✓Final answer

\dfrac1{4}\log\left|\dfrac{2+x}{2-x}\right|+c

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Q.Find the integrals of the following:

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