Una integral quasi immediata és una integral de la forma:
\(\displaystyle \int f(u(x)) \cdot u'(x) \ \mathrm{d}x\)
on \(u(x)\) és una funció de derivada \(u'(x)\) i \(f(x)\) és una funció de la qual es coneix una funció primitiva \(F(x)\). Aleshores aplicant la regla de la cadena en sentit contrari:
\(\bbox[15px,border:1px solid]{ \displaystyle \int f(u(x)) \cdot u'(x) \ \mathrm{d}x = F(u(x))+C }\)
Això permet aplicar a més funcions la taula d'integrals immediates de l'apartat anterior.
| Integral quasi immediata | Integral immediata associada |
| \(\displaystyle \int \big[u(x)\big]^n \cdot u'(x) \,\mathrm{d}x = \frac{\big[u(x)\big]^{n+1}}{n+1} + C \quad\quad (n \ne -1)\) | \(\displaystyle \int x^n \,\mathrm{d}x = \frac{x^{n+1}}{n+1} + C \quad\quad (n \ne -1)\) |
| \(\displaystyle \int \frac{u'(x)}{u(x)} \,\mathrm{d}x = \ln \left\lvert u(x) \right\rvert + C \) | \(\displaystyle \int \frac{1}{x} \,\mathrm{d}x = \ln \left\lvert x \right\rvert + C \) |
Exemples
Exercici 7
Calcula les següents integrals indefinides:
| a) \(\displaystyle \int \left( \sin x +4 \right)^5 \, \cos x \,\mathrm{d}x \) | Solució: | |
| b) \(\displaystyle \int \left( x+3 \right)^2 \,\mathrm{d}x \) | Solució: | |
| c) \(\displaystyle \int \frac{x-5}{x^2-10x+21}\mathrm{d}x \) | Solució: | |
| d) \(\displaystyle \int \sqrt[3]{x^2+x}\,\left( 2x+1 \right) \,\mathrm{d}x \) | Solució: | |
| e) \(\displaystyle \int \sin^4 x\cos x\,\mathrm{d}x \) | Solució: | |
| f) \(\displaystyle \int \tan{x}\,\mathrm{d}x \) | Solució: | |
| g) \(\displaystyle \int \sin^3 x \,\mathrm{d}x \) | Solució: | |
| h) \(\displaystyle \int \frac{\ln x}{x} \,\mathrm{d}x \) | Solució: |
| Integral quasi immediata | Integral immediata associada |
| \(\displaystyle \int \mathrm{e}^{u(x)} \cdot u'(x)\,\mathrm{d}x = \mathrm{e}^{u(x)} + C \) | \(\displaystyle \int \mathrm{e}^x \,\mathrm{d}x = \mathrm{e}^x + C \) |
| \(\displaystyle \int a^{u(x)} \cdot u'(x)\,\mathrm{d}x = \frac{a^{u(x)}}{\ln a} + C \) | \(\displaystyle \int a^x \,\mathrm{d}x = \frac{a^x}{\ln a} + C \) |
Exemples
Exercici 8
Calcula les següents integrals indefinides:
| a) \(\displaystyle \int 2^x \,\mathrm{d}x \) | Solució: | |
| b) \(\displaystyle \int \mathrm{e}^{x^2+2x} \cdot \left( x+1 \right) \,\mathrm{d}x \) | Solució: | |
| c) \(\displaystyle \int \frac{\mathrm{e}^{\tan x}}{\cos^2 x} \,\mathrm{d}x \) | Solució: |
| Integral quasi immediata | Integral immediata associada |
| \(\displaystyle \int \sin \big[u(x)\big] \cdot u'(x) \,\mathrm{d}x = -\cos \big[u(x)\big] + C \) | \(\displaystyle \int \sin x \,\mathrm{d}x = -\cos x + C \) |
| \(\displaystyle \int \cos \big[u(x)\big] \cdot u'(x) \,\mathrm{d}x = \sin \big[u(x)\big] + C \) | \(\displaystyle \int \cos x \,\mathrm{d}x = \sin x + C \) |
| \(\displaystyle \int \frac{u'(x)}{\cos^2 \big[u(x)\big]} \,\mathrm{d}x = \tan x + C \) | \(\displaystyle \int \frac{1}{\cos^2 x} \,\mathrm{d}x = \tan x + C \) |
| \(\displaystyle \int \left( 1+\tan^2 \big[u(x)\big] \right) \cdot u'(x)\,\mathrm{d}x = \tan x + C \) | \(\displaystyle \int \left( 1+\tan^2 x \right) \,\mathrm{d}x = \tan x + C \) |
Exemples
Exercici 9
Calcula les següents integrals indefinides:
| a) \(\displaystyle \int \frac{\sin\sqrt{x}}{\sqrt{x}} \,\mathrm{d}x \) | Solució: | |
| b) \(\displaystyle \int \frac{x}{\cos^2 \left( 2+x^2 \right)} \,\mathrm{d}x \) | Solució: | |
| c) \(\displaystyle \int x \cdot \cos \left( x^2+5 \right) \,\mathrm{d}x \) | Solució: |
| Integral quasi immediata | Integral immediata associada |
| \(\displaystyle \int \frac{u'(x)}{\sqrt{1-\big[u(x)\big]^2}} \,\mathrm{d}x = \left\lbrace\begin{array}{r} \arcsin \big[u(x)\big] + C \\[6pt] -\arccos \big[u(x)\big] + C \end{array}\right.\) | \(\displaystyle \int \frac{1}{\sqrt{1-x^2}} \,\mathrm{d}x = \left\lbrace\begin{array}{r} \arcsin x + C \\[6pt] -\arccos x + C \end{array}\right.\) |
| \(\displaystyle \int \frac{u'(x)}{1+\big[u(x)\big]^2} \,\mathrm{d}x = \arctan \big[u(x)\big] + C \) | \(\displaystyle \int \frac{1}{1+x^2} \,\mathrm{d}x = \arctan x + C \) |
Exemples