aufgabe 2b

$$\begin{gathered} {\int }_a^b\underset{t}{\underbrace{(3x-5)}}{ \\ }^{11}dx \end{gathered}$$

Substitution ohne evidente innere Ableitung, Nebenrechnung:

$$\begin{array}{rl}t& =3x-5\\ t+5& =3x\\ \frac{t+5}{3}& =x\end{array}$$

Daraus folgt

$$\begin{array}{rl}\frac{dx}{dt}& =\frac{1}{3}\\ ⟺dx& =\frac{dt}{3}\end{array}$$

Substituiere:

$${\int }_{t(a)}^{t(b)}{t}^{11}\cdot \frac{1}{3}dt$$

Und rechne aus:

$$\begin{array}{rl}\dots & =\frac{1}{3}{\int }_{t(a)}^{t(b)}{t}^{11}dt\\ & =\frac{1}{3}{\left[\frac{1}{12}{t}^{12}\right]}_{t(a)}^{t(b)}\\ & =\frac{1}{3}\cdot \left(\frac{1}{12}(t(b){)}^{12}-\frac{1}{12}(t(a){)}^{12}\right)\\ & =\frac{1}{3\cdot 12}((3b-5{)}^{12}-(3a-5{)}^{12})\\ & =\frac{1}{36}((3b-5{)}^{12}-(3a-5{)}^{12})\end{array}$$