Cos x'in İntegrali

Cos x'in İntegrali Nedir ?

Cos x'in integrali sin x+c'dir.

$\int cosxdx=sinx+c$

Cos x'in İntegralini Bulma

$\int cosxdx=?$

$cosx=u$

$d(cosx)=du$

$(cosx{)}^{\prime }dx=du$

$\boxed{(cosx{)}^{\prime }=-sinx}$

$-sinxdx=du$

$dx=-\frac{du}{sinx}$

$cosx=u$

$sinx=\sqrt{1-u^2}$

$dx=-\frac{du}{\sqrt{1-u^2}}$

$\int cosxdx=\int u.-\frac{du}{\sqrt{1-u^2}}$

$\int cosxdx=\int -\frac{udu}{\sqrt{1-u^2}}$

$\sqrt{1-u^2}=v$

$(\sqrt{1-u^2}{)}^2=v^2$

$\sqrt{(1-u^2{)}^2}=v^2$

$1-u^2=v^2$

$d(1-u^2)=d(v^2)$

$(1-u^2{)}^{\prime }du=(v^2{)}^{\prime }dv$

$-\cancel{2}udu=\cancel{2}vdv$

$-udu=vdv$

$\int cosxdx=\int \frac{\cancel{v}dv}{\cancel{v}}$

$\int cosxdx=\int dv$

$\int cosxdx=v+c$

$\int cosxdx=\sqrt{1-u^2}+c$

$\int cosxdx=sinx+c$