Calcolatore di integrali definiti e impropri
Calcola integrali definiti e impropri passo dopo passo
Il calcolatore cercherà di valutare l'integrale definito (cioè con estremi), inclusi quelli impropri, mostrando i passaggi.
Solution
Your input: calculate $$$\int_{0}^{6}\left( 38 \left(\frac{6}{5}\right)^{t} \right)dt$$$
First, calculate the corresponding indefinite integral: $$$\int{38 \left(\frac{6}{5}\right)^{t} d t}=\frac{38 \left(\frac{6}{5}\right)^{t}}{- \ln{\left(5 \right)} + \ln{\left(6 \right)}}$$$ (for steps, see indefinite integral calculator)
According to the Fundamental Theorem of Calculus, $$$\int_a^b F(x) dx=f(b)-f(a)$$$, so just evaluate the integral at the endpoints, and that's the answer.
$$$\left(\frac{38 \left(\frac{6}{5}\right)^{t}}{- \ln{\left(5 \right)} + \ln{\left(6 \right)}}\right)|_{\left(t=6\right)}=\frac{1772928}{15625 \left(- \ln{\left(5 \right)} + \ln{\left(6 \right)}\right)}$$$
$$$\left(\frac{38 \left(\frac{6}{5}\right)^{t}}{- \ln{\left(5 \right)} + \ln{\left(6 \right)}}\right)|_{\left(t=0\right)}=\frac{38}{- \ln{\left(5 \right)} + \ln{\left(6 \right)}}$$$
$$$\int_{0}^{6}\left( 38 \left(\frac{6}{5}\right)^{t} \right)dt=\left(\frac{38 \left(\frac{6}{5}\right)^{t}}{- \ln{\left(5 \right)} + \ln{\left(6 \right)}}\right)|_{\left(t=6\right)}-\left(\frac{38 \left(\frac{6}{5}\right)^{t}}{- \ln{\left(5 \right)} + \ln{\left(6 \right)}}\right)|_{\left(t=0\right)}=\frac{1179178}{15625 \left(- \ln{\left(5 \right)} + \ln{\left(6 \right)}\right)}$$$
Answer: $$$\int_{0}^{6}\left( 38 \left(\frac{6}{5}\right)^{t} \right)dt=\frac{1179178}{15625 \left(- \ln{\left(5 \right)} + \ln{\left(6 \right)}\right)}\approx 413.924679709088$$$