Przydatne wzory pochodne,całki
Zad.1
$$\int \frac{e^{\sqrt{x}}}{\sqrt{x}} dx= \left|\left| u = \sqrt{x} , x = u^2,dx = 2udu\right| \right|=\int 2 e^u du = 2 e^u +C =2e^{\sqrt{x}} + C $$
Zad.2
$$ \int \sin(\ln x) dx =\left|\left| u = \ln x , x = e^u , dx = e^u du \right| \right|= \int e^u \sin u du $$
Teraz całkujemy przez części.$$\int e^u \sin u du=\left|\left| f=e^u ,g’ =sinu , f’=e^u, g=-cosu \right| \right|= -e^u \cos u + \int e^u \cos u du =$$
$$\left|\left| f=e^u ,g’ =cosu , f’=e^u, g=sinu \right| \right|=-e^u \cos u +e^u sinu – \int e^u sinu du$$
$$ \int e^u \sin u du=\frac{-e^u \cos u + e^u \sin u }{2}+ C=||u = \ln x ||=\frac{-x \cos(\ln x) + x \sin(\ln x)}{2}+ C$$
Zad.3
$$ \int \frac{dx}{x\ln x}= \left|\left| u = \ln x,du = \frac{dx}{x} \right| \right|=\int \frac{1}{u} du = \ln |u| + C= \ln |\ln x| + C$$
Zad.4
$$ \int x^2 \cos(x^3) dx=\left|\left| u = x^3 , du = 3x^2 dx\right| \right|= \frac{1}{3} \int \cos u du=\frac{1}{3} \sin u + C = \frac{1}{3} \sin(x^3) + C $$
Zad.5
$$ \int \frac{\sin \sqrt{x}}{\sqrt{x}} dx=\left|\left| u = \sqrt{x} , x = u^2,dx = 2udu \right| \right|= 2 \int \sin u du= -2cosu+C=-2cos \sqrt{x}+C $$
Zad.6
$$ \int \frac{xdx}{(x^2 + 1)^2} = \left|\left| u = x^2 + 1 , du = 2x dx \right| \right| =\frac{1}{2} \int \frac{du}{u^2}= -\frac{1}{2u} + C = -\frac{1}{2(x^2 + 1)} + C $$
Zad.7
$$ \int e^{5x} \sin(2e^{5x}) dx =\left|\left| u =2e^{5x}, du =10e^{5x} dx \right| \right| =$$$$=\frac{1}{10} \int \sin u du=-\frac{1}{10} \cos u + C = -\frac{1}{10} \cos(2e^{5x}) + C $$
Zad.8
$$ \int \frac{x^{11}}{\sqrt{x^4 – 1}} dx =||u=x^4 – 1,du=4x^3dx, x^4=u+1||=\frac{1}{4}\int\frac{(u+1)^2 }{\sqrt{u}}du=$$$$=\frac{1}{4} \int \frac{u^2}{\sqrt{u}} + \frac{2u}{\sqrt{u}}+ \frac{1}{\sqrt{u}}du=\frac{1}{4} \int u^{3/2} + 2\sqrt{u}+ u^{-1/2}du=\frac{1}{4} (\frac{2}{5} u^{5/2} + \frac{4}{3} u^{3/2} + 2u^{1/2} ) =$$$$=\frac{1}{10}(x^4-1)^{5/2}+\frac{1}{3}(x^4-1)^{3/2}+\frac{1}{2}(x^4-1)^{1/2} +C$$
Zad.9
$$ \int \frac{\ln ^3x}{x} dx =\left|\left| u = \ln x, du = \frac{dx}{x} \right| \right| = \int u^3 du = \frac{u^4}{4} + C = \frac{(\ln x)^4}{4} + C $$
Zad.10
$$ \int \frac{e^{2x}}{e^x + 1} dx=|| u=e^x + 1,e^x=u-1, du = e^x dx||= $$$$=\int \frac{(u-1)du}{u} = u-ln |u| + C=e^x – \ln (e^x + 1)+ C $$
Zad.11
$$ \int \frac{sinxdx}{(cos x)^2} =||u = cosx, du = -sinxdx||=2 \int \frac{-du}{u^2} =\frac{-2}{u} +C=\frac{-2}{cosx} +C$$
Zad.12
$$ \int e^x \sqrt{1 + e^x} dx =||u=1+e^x, du=e^xdx||=\int\sqrt{u} du = \frac{2}{3}\sqrt{u^3} +C=\frac{2}{3}\sqrt{(1+e^x)^3} +C $$
Zad.13
$$ \int \frac{x^3dx}{(6 + x^2)^2} =|| u = x^2 + 6,x^2=u – 6, du = 2x dx ||=$$$$= \frac{1}{2} \int \frac{(u – 6)du}{u^2}=
\frac{1}{2} \int \frac{-6du}{u^2} +\frac{1}{2} \int \frac{udu}{u^2}= \frac{3}{u} + \frac{1}{2}lnu +C=$$$$=\frac{3}{x^2 +6} + \frac{1}{2}ln(x^2+6) + C$$
Zad.14
$$ \int \frac{3arctan x}{x^2+1} dx =||u = \arctan x, du = \frac{dx}{1 + x^2}||= \int 3u du = \frac{3u^2}{2} + C =\frac{3(\arctan x)^2}{2} + C $$