El blog de Carlos: Integrales

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La integral de cero es igual a la constante $C$ .

${ \int 0 dx = C }$

Integral de una potencia

La integral de una potencia es igual a la variable elevada a la potencia $n+1$ sobre $n+1$ sumando una constante.

$\displaystyle { \int x^n dx = \frac{x^{n+1}}{n+1} + C \qquad\quad n\neq -1}$

$\displaystyle { \int u^n \cdot u^{'} du = \frac{u^{n+1}}{n+1} + C \qquad\quad n\neq -1}$

Ejemplos de integrales

1 ${ \int 7 dx }$

${ \int 7 dx = 7x + C }$

2 ${ \int x^6 dx }$

$\displaystyle { \int x^6 dx = \frac{x^{6+1}}{6+1} + C = \frac{x^7}{7} + C }$

3 ${ \int 7 x^3 dx }$

$\displaystyle { \int 7 x^3 dx = \frac{7 x^{3+1}}{3+1} + C = \frac{7 x^4}{4} + C }$

4 $\displaystyle { \int x^{\frac{2}{3}} dx }$

$\displaystyle { \int x^{\frac{2}{3}} dx = \frac{x^{\frac{2}{3}+1}}{\frac{2}{3}+1} + C = \frac{x^{\frac{5}{3}}}{\frac{5}{3}} + C = \frac{3 \sqrt[3]{x^5}}{5} + C = \frac{3 \sqrt[3]{x^3\cdot x^2}}{5} + C = \frac{3 x\cdot \sqrt[3]{x^2}}{5} + C}$

5 $\displaystyle { \int \frac{3}{x^4} dx }$

$\displaystyle { \int \frac{3}{x^4} dx = \int 3x^{-4} dx = \frac{3x^{-4+1}}{-4+1} + C = \frac{3x^{-3}}{-3} + C = -x^{-3} + C = -\frac{1}{x^{3}} + C }$

6 ${ \int \sqrt[3]{x} dx }$

$\displaystyle { \int \sqrt[3]{x} dx = \int x^{\frac{1}{3}} dx = \frac{x^{\frac{1}{3}+1}}{\frac{1}{3}+1} + C = \frac{x^{\frac{4}{3}}}{{\frac{4}{3}}} + C = \frac{3}{4}x^{\frac{4}{3}} + C =\frac{3}{4}x\sqrt[3]{x} + C }$

7 $\displaystyle { \int \frac{1}{\sqrt[4]{x}} dx }$

$\displaystyle { \int \frac{1}{\sqrt[4]{x}} dx = \int x^{\frac{-1}{4}} dx = \frac{x^{\frac{-1}{4}+1}}{\frac{-1}{4}+1} + C = \frac{x^{\frac{3}{4}}}{\frac{3}{4}}} + C = \frac{4}{3}x^{\frac{3}{4}} + C =\frac{4}{3}\sqrt[4]{x^3} + C}$

8 $\displaystyle { \int \frac{1}{\sqrt[3]{x^2}} dx }$

$\displaystyle { \int \frac{1}{\sqrt[3]{x^2}} dx = \int x^{\frac{-2}{3}} dx = \frac{x^{\frac{-2}{3}+1}}{\frac{-2}{3}+1}}+ C = \frac{x^{\frac{1}{3}}}{\frac{1}{3}} + C = 3\sqrt[3]{x} + C }$

9 $\displaystyle { \int \frac{1}{x^2 \sqrt[5]{x^2}} dx }$

$\displaystyle { \int \frac{1}{x^2 \sqrt[5]{x^2}} dx = \int x^{-2} x^{\frac{-2}{5}} dx = \int x^{\frac{-12}{5}} dx = \frac{x^{\frac{-7}{5}}}{\frac{-7}{5}} + C = -\frac{5}{7 \sqrt[5]{x^7}} +C }$

10 ${ \int (x^4 - 6x^2 -2x + 4) dx }$

$\displaystyle { \int (x^4 - 6x^2 -2x + 4) dx = \frac{x^5}{5} - \frac{6x^3}{3} -\frac{2x^2}{2} + 4x + C = \frac{x^5}{5} - 2x^3 - x^2 + 4x + C }$

11 $\displaystyle { \int (3\sqrt{x} + \frac{10}{x^6}) dx }$

$\displaystyle { \int (3\sqrt{x} + \frac{10}{x^6}) dx = \int (3x^{\frac{1}{2}} + 10x^{-6}) dx = \frac{3x^{\frac{3}{2}}}{\frac{3}{2}} - \frac{10 x^{-5}}{5} + C = 2x\sqrt{x} - \frac{2}{x^5} + C }$

12 $\displaystyle { \int (\frac{x^2 + \sqrt[3]{x^2}}{\sqrt{x}}) dx }$

$\displaystyle { \int (\frac{x^2 + \sqrt[3]{x^2}}{\sqrt{x}}) dx = \int (\frac{x^2}{\sqrt{x}} + \frac{\sqrt[3]{x^2}}{\sqrt{x}}) dx = \int (x^{\frac{3}{2}} + x^{\frac{1}{6}}) dx = }$

$\displaystyle { = \frac{x^{\frac{5}{2}}}{\frac{5}{2}} + \frac{x^{\frac{7}{6}}}{\frac{7}{6}} + C = \frac{2\sqrt{x^5}}{5} + \frac{6\sqrt[6]{x^7}}{7} +C = \frac{2x^2\sqrt{x}}{5} + \frac{6x\sqrt[6]{x}}{7} + C }$

13 $\displaystyle { \int (\sqrt{5x} + \sqrt{\frac{5}{x}}) dx }$

$\displaystyle { \int (\sqrt{5x} + \sqrt{\frac{5}{x}}) dx = \int (\sqrt{5}x^{\frac{1}{2}} +\sqrt{5}x^{\frac{-1}{2}}) dx = \sqrt{5}\frac{x^{\frac{3}{2}}}{\frac{3}{2}} + \sqrt{5}\frac{x^{\frac{1}{2}}}{\frac{1}{2}} + C = }$

$\displaystyle { \frac{2\sqrt{5}x\sqrt{x}}{3} + 2\sqrt{5}\sqrt{x} + C = \frac{2x\sqrt{5x}}{3} + 2\sqrt{5x} + C }$

14 $\displaystyle { \int (\frac{3\sqrt{x}- 5\sqrt[3]{x^2}}{2\sqrt[4]{x}}) dx }$

$\displaystyle { \int (\frac{3\sqrt{x}- 5\sqrt[3]{x^2}}{2\sqrt[4]{x}}) dx = \int (\frac{3\sqrt{x}}{2\sqrt[4]{x}} - \frac{5\sqrt[3]{x^2}}{2\sqrt[4]{x}}) dx = \int (\frac{3}{2}x^{\frac{1}{4}} - \frac{5}{2}x^{\frac{5}{12}}) dx = }$

$\displaystyle { = \frac{3}{2}\frac{x^{\frac{5}{4}}}{\frac{5}{4}} - \frac{5}{2}\frac{x^{\frac{17}{12}}}{\frac{17}{12}} + C = \frac{6}{5}\sqrt[4]{x^5} - \frac{30}{17}\sqrt[12]{x^{17}} + C}$

15 ${ \int \sin x \cos x dx }$

$\displaystyle { \int \sin x \cos x dx = \frac{1}{2}\sin^2 x + C }$

16 $\displaystyle { \int \sin^2 \frac{x}{2} \cos \frac{x}{2} dx }$

$\displaystyle { \int \sin^2 \frac{x}{2} \cos \frac{x}{2} dx = 2 \int \sin^2 \frac{x}{2} \cos \frac{x}{2} \cdot \frac{1}{2} dx = \frac{2}{3}\sin^{3} (\frac{x}{2}) +C }$

17 ${ \int (\tan^3 x + \tan^5 x) dx }$

$\displaystyle { \int (\tan^3 x + \tan^5 x) dx = \int \tan^3 x (1+ \tan^2 x) dx = \frac{1}{4}\tan^4 x + C }$

18 ${ \int \sec^2 x \sqrt{\tan x} dx }$

$\displaystyle { \int \sec^2 x \sqrt{\tan x} dx = \int \sec^2 x (\tan x)^{\frac{1}{2}} dx = \frac{(\tan x)^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{3}\sqrt{\tan^3 x} +C }$

19 ${ \int \cot x \sqrt{In \sin x} dx }$

$\displaystyle { \int \cot x \sqrt{In \sin x} dx = \int\cot x (In \sin x)^{\frac{1}{2}} dx = \frac{(In \sin x)^{\frac{3}{2}}}{\frac{3}{2}} + C = \frac{2}{3}\sqrt{(In \sin x)^3} + C }$

20 $\displaystyle { \int \frac{\sin 3x}{\sqrt{2 + \cos 3x}} dx }$

$\displaystyle { \int \frac{\sin 3x}{\sqrt{2 + \cos 3x}} dx = -\frac{1}{3} \int (2 + \cos 3x)^{\frac{-1}{2}} \sin 3x (-3) dx = -\frac{2}{3} \sqrt{2+\cos 3x} + C }$

https://www.superprof.es/apuntes/escolar/matematicas/calculo/integrales/integrales-inmediatas.html#tema_integral-de-una-constante

El blog de Carlos

viernes, 21 de enero de 2022

Integrales

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Integral de una potencia

Ejemplos de integrales

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