Całki oznaczone

[Robson1416]

Mam problem z tymi całkami:

zad 1 - tutaj mam policzyć je przed podstawianie

\(\int_{0}^{ \frac{1}{2} ln 3} \frac{e^{x}dx}{1+e^{2x}}\)


\(\int_{0}^{ \pi} sinxe^{cos x}\)


\(\int_{1}^{3} \frac{xdx}{ \sqrt{x+1} }\)


\(\int_{\frac{1}{4}}^{1} \frac{dx}{ \sqrt{x}(4-x) }\)

[domino21]

\(\int_0^{\pi} \sin x e^{\cos x} dx= (*)\)

\(\int \sin x e^{\cos x} dx =\left(\cos x=t \\ -\sin x dx =dt \\ \sin x dx=-dt \right)=-\int e^t dx =-e^t+C=-e^{\cos x} +C\)

\((*)=-e^{\cos x} |_0^{\pi} =-e^{\cos \pi} +e^{\cos 0} =-e^{-1} +e^1=e-\frac{1}{e}\)

[domino21]

\(\int_1^3 \frac{ x dx}{\sqrt{x+1}}=(*)\)

\(\int \frac{ x dx}{\sqrt{x+1}}=\left( \sqrt{x+1} =t \\x+1=t^2 \\ dx =2t dt \\ x=t^2-1 \right)=2\int \frac{ t^2-1}{t}t dt=2\int t^2-1 dt=\frac{2}{3} t^3-2t+C=\frac{2}{3} (x+1)\sqrt{x+1} -2\sqrt{x+1} +C=\)

\(=\frac{2}{3} (x-2)\sqrt{x+1}+C\)

\((*)=\frac{2}{3} (x-2)\sqrt{x+1} |_1^3 =\frac{2}{3}(2+\sqrt{2})\)

[domino21]

\(\int _0 ^{\frac{1}{2}\ln 3} \frac{e^x }{1+e^{2x}} dx=(*)\)

\(\int \frac{e^x}{1+e^{2x}} dx =\left(e^x =t \\ e^x dx=dt \right) =\int \frac{dt}{1+t^2}=arc tg t+C =arc tg e^x +C\)

\((*)=arc tg e^x | _0 ^{\frac{1}{2} \ln 3} =\frac{\pi}{3} -\frac{\pi}{4} =\frac{\pi}{12}\)