6a)
\( \int \limits_{1}^{2} x^{2}+\frac{1}{x} d x =[\frac{1}{3} x^3 +\ln(|x|) ]_{1}^{2} =[\frac{8}{3} +\ln(2) ]-[ \frac{1}{3} +\ln(1) ]=\frac{7}{3}+\ln(2)\) wobei \(\ln(1)=0\)
7)
\( \int \limits_{0}^{q} \sqrt{q-x} d x, \quad q \in I R, q>0 \)
Substitution:
\(q-x=z\) → \(x=q-z\)
\( \frac{dx}{dz}=-1 \) \( dz=-dz \)
Grenzen abändern:
untere Grenze: \(x=0\)→ \(z=q\)
obere Grenze: \(x=q\)→ \(z=0\)
\( \int\limits_{q}^{0}-\sqrt{z}dz=-\int\limits_{q}^{0}z^{\frac{1}{2}}dz=[-\frac{z^{\frac{1}{2}+1}}{\frac{3}{2}}]_{q}^{0}=[-\frac{2}{3}z^{\frac{3}{2}}]_{q}^{0}=[0]-[-\frac{2}{3}q^{\frac{3}{2}}]=\frac{2}{3}q^{\frac{3}{2}}\)
