## 140. Transcendental Functions.

Owing to the immense variety of the different classes of transcendental functions, the theory of their integration is a good deal less systematic than that of the integration of rational or algebraical functions. We shall consider in order a few classes of transcendental functions whose integrals can always be found.

## 141. Polynomials in cosines and sines of multiples of \(x\).

We can always integrate any function which is the sum of a finite number of terms such as \[A\cos^{m} ax \sin^{m’} ax \cos^{n} bx \sin^{n’} bx\dots,\] where \(m\), \(m’\), \(n\), \(n’\), … are positive integers and \(a\), \(b\), … any real numbers whatever. For such a term can be expressed as the sum of a finite number of terms of the types \[\alpha\cos\{(pa + qb + \dots)x\},\quad \beta \sin\{(pa + qb + \dots)x\}\] and the integrals of these terms can be written down at once.

## 142. The integrals \(\int x^{n}\cos x\, dx\), \(\int x^{n}\sin x\, dx\) and associated integrals.

The method of integration by parts enables us to generalise the preceding results. For \[\begin{aligned} \int x^{n}\cos x\, dx &= & &x^{n}\sin x &&- n\int x^{n-1}\sin x\, dx,\\ \int x^{n}\sin x\, dx &= &-&x^{n}\cos x &&+ n\int x^{n-1}\cos x\, dx,\end{aligned}\] and clearly the integrals can be calculated completely by a repetition of this process whenever \(n\) is a positive integer. It follows that we can always calculate \(\int x^{n}\cos ax\, dx\) and \(\int x^{n}\sin ax\, dx\) if \(n\) is a positive integer; and so, by a process similar to that of the preceding paragraph, we can calculate \[\int P(x, \cos ax, \sin ax, \cos bx, \sin bx, \dots)\, dx,\] where \(P\) is any polynomial.

## 143. Rational Functions of \(\cos x\) and \(\sin x\).

The integral of any rational function of \(\cos x\) and \(\sin x\) may be calculated by the substitution \(\tan \frac{1}{2}x = t\). For \[\cos x = \frac{1 – t^{2}}{1 + t^{2}},\quad \sin x = \frac{2t}{1 + t^{2}},\quad \frac{dx}{dt} = \frac{2}{1 + t^{2}},\] so that the substitution reduces the integral to that of a rational function of \(t\).

## 144. Integrals involving \(\arcsin x\), \(\arctan x\), and \(\log x\).

The integrals of the inverse sine and tangent and of the logarithm can easily be calculated by integration by parts. Thus \[\begin{aligned} \int \arcsin x\, dx &= x\arcsin x – \int \frac{x\, dx}{\sqrt{1 – x^{2}}} = x\arcsin x + \sqrt{1 – x^{2}},\\ \int \arctan x\, dx &= x\arctan x – \int \frac{x\, dx}{1 + x^{2}} = x\arctan x – \tfrac{1}{2} \log(1 + x^{2}),\\ \int \log x\, dx &= x\log x – \int dx = x(\log x – 1).\end{aligned}\]

It is easy to see that if we can find the integral of \(y = f(x)\) then we can always find that of \(x = \phi(y)\), where \(\phi\) is the function inverse to \(f\). For on making the substitution \(y = f(x)\) we obtain \[\int \phi(y)\, dy = \int xf'(x)\, dx = xf(x) – \int f(x)\, dx.\] The reader should evaluate the integrals of \(\arcsin y\) and \(\arctan y\) in this way.

Integrals of the form \[\int P(x, \arcsin x)\, dx,\quad \int P(x, \log x)\, dx,\] where \(P\) is a polynomial, can always be calculated. Take the first form, for example. We have to calculate a number of integrals of the type \(\int x^{m} (\arcsin x)^{n}\, dx\). Making the substitution \(x = \sin y\), we obtain \(\int y^{n}\sin^{m}y \cos y\, dy\), which can be found by the method of § 142. In the case of the second form we have to calculate a number of integrals of the type \(\int x^{m} (\log x)^{n}\, dx\). Integrating by parts we obtain \[\int x^{m}(\log x)^{n}\, dx = \frac{x^{m+1} (\log x)^{n}}{m + 1} – \frac{n}{m + 1} \int x^{m}(\log x)^{n-1}\, dx,\] and it is evident that by repeating this process often enough we shall always arrive finally at the complete value of the integral.

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