mtk: Integral Trigonometri

Senin, 03 Januari 2011

Integral Trigonometri

-Integral cos x
misal f(x)=sin x

$f(x)=sinx\rightarrow \frac{df(x)}{dx}=cosx\rightarrow df(x)=cosxdx\rightarrow$

$\int df(x)=\int cosxdx\rightarrow \int cosxdx=\int df(x)=f(x)+C=sinx+C$

-Integral sin x
misal f(x)=cos x

$f(x)=cosx\rightarrow \frac{df(x)}{dx}=-sinx\rightarrow df(x)=-sinxdx\rightarrow$

$\int df(x)=-\int sinxdx\rightarrow -\int sinxdx=\int df(x)=f(x)+C=cosx+C$
$\int sinxdx=-cosx+C$

-Integral Tan x
misal:
u=cos x, maka du=-sin x dx

$\int tanxdx=\int \frac{sinx}{cosx}dx=\int \frac{sinxdx}{cosx}=\int \frac{-du}{u}$

$\int tanxdx=-ln|u|+C=-ln|cosx|+C$

-Integral sec x
misal:
u=sin x,maka du=cos dx

$\int secxdx=\int \frac{1}{cosx}dx=\int \frac{1}{cosx}.\frac{cosx}{cosx}dx=\int \frac{cosx}{cosx^2x}dx=\int \frac{cosx}{1-sin^2x}dx\rightarrow$

$\int secxdx=\int \frac{cosxdx}{1-sin^2x}=\int \frac{du}{1-u^2}=\frac{1}{2}\int (\frac{1}{1+u}+\frac{1}{1-u})du\rightarrow$

$\int secxdx=\frac{1}{2}(ln|1+u|-ln|1-u|)+C=\frac{1}{2}ln|\frac{1+u}{1-u}|+C\rightarrow$

$\int secxdx=\frac{1}{2}ln|\frac{1+sinx}{1-sinx}|+C$

-Integral cosec x
Misal:
u=cos ,maka du=-sin xdx

$\int cosecxdx=\int \frac{1}{sinx}dx=\int \frac{1}{sinx}.\frac{sinx}{sinx}dx=\int \frac{sinx}{sin^2x}dx=\int \frac{sinx}{1-cos^2x}dx\rightarrow$

$\int cosecxdx=\int \frac{sinxdx}{1-cos^2x}=\int\frac{-du}{1-u^2}=-\frac{1}{2}\int(\frac{1}{1+u}+\frac{1}{1-u})du\rightarrow$

$\int cosecxdx=-\frac{1}{2}(ln|1+u|-ln|1-u|)+C=-\frac{1}{2}ln|\frac{1+u}{1-u}|+C\rightarrow$

$\int cosecxdx=-\frac{1}{2}ln|\frac{1+cosx}{1-cosx}|+C$

-Integral cotan x
Misal:
u=sin x, maka du=cosx dx

$\int cotan xdx=\int \frac{cosx}{sinx}dx=\int \frac{du}{u}=ln|u|+C\rightarrow$

$\int cotan xdx=ln|sinx|+C$

1 komentar:

Husni Syafnur5 Maret 2011 pukul 06.17
wachhh,,, Postingnya Mantap,,,
Pas bangett ada tugas kuliah ini,,,,(^_^),,

Boleh ya saya copy,,,,
BalasHapus
Balasan

Tambahkan komentar

mtk

Total Tayangan Halaman

Senin, 03 Januari 2011

Integral Trigonometri

1 komentar:

berapakah x.......?

Arsip Blog

Cari Blog Ini