# Integration by Parts via Tabular Integration, where am I wrong?

Evaluate $\displaystyle \frac{1}{5000}\int_0^{100} t (1.1)^t \mathrm d t$.

What am I doing wrong with tabular integration for this problem?

Im using $t$ for the derivative term (so it ends up as $t\to1\to0)$ and integrating with $(1.1)^t$ so the first term is: $\displaystyle \frac{(1.1)^{t+1}}{t+1}$, and the second is $\displaystyle \frac{(1.1)^{t+2}}{(t+2)(t+1)}$.

Combining terms and solving gives me approximately $3000$ which is wrong.

• Have you taken care of the $\pm$ signs to the left of the tabular "environment"? Dec 1 '12 at 2:25

You are using a false integration formula to integrate $1.1^t$. There is a big difference between a function of the shape $t^k$, where $k$ is a constant, and a function of the shape $a^t$, where $a$ is a constant.
Except for the case $k=-1$, $\dfrac{1}{k+1}t^{k+1}$ is an indefinite integral of $t^k$. However, this formula does not apply when the variable is in the exponent.
We now proceed to find an antiderivative of $1.1^t$. Note that $1.1^t=e^{(\ln(1.1) t}$. Integrate. By using the substitution $u=(\ln(1.1))t$, or otherwise, we get that one indefinite integral is $\dfrac{1}{\ln(1.1)}e^{(\ln(1.1)) t}$. This can be rewritten as $\dfrac{1}{\ln(1.1)}(1.1^t)$.