Use substitution to find the indefinite integral. \[ \int(1-t) e^{26 t-13 t^{2}} d t \] Describe the most appropriate substitution case and the values of $u$ and du. Select the correct choice below and fill in the answer boxes within your choice. A. Substitute $u$ for the quantity in the denominator. Let $u=\square$, so that $d u=(\quad) d t$. B. Substitute $\mathrm{u}$ for the exponent on $e$. Let $\mathrm{u}=\square$, so that $\mathrm{du}=(\square \mathrm{dt}$. C. Substitute $u$ for the quantity under a root or raised to a power. Let $u=\square$, so that du $=($ ) $d t$

Step 1 ：The most appropriate substitution case is to substitute $u$ for the exponent on $e$. So, let $u=26t-13t^{2}$.

Step 2 ：To find $du$, we differentiate $u$ with respect to $t$. So, $du=(26-26t)dt$.

Step 3 ：Substitute $u$ and $du$ into the integral, we get $\int(1-t)e^{u}(26-26t)dt$.

Step 4 ：Simplify the integral, we get $26\int e^{u}dt - 26\int te^{u}dt$.

Step 5 ：The integral of $e^{u}$ with respect to $u$ is $e^{u}$, and the integral of $te^{u}$ with respect to $u$ is $-e^{u}$ by integration by parts.

Step 6 ：So, the integral becomes $26e^{u} + 26e^{u}$.

Step 7 ：Substitute $u$ back into the integral, we get $26e^{26t-13t^{2}} + 26e^{26t-13t^{2}}$.

Step 8 ：Simplify the expression, we get $52e^{26t-13t^{2}}$.

Step 9 ：So, the indefinite integral of $(1-t)e^{26t-13t^{2}}$ with respect to $t$ is $52e^{26t-13t^{2}} + C$, where $C$ is the constant of integration.

Step 10 ：Check the result by differentiating it with respect to $t$, we get $(1-t)e^{26t-13t^{2}}$, which is the original integrand. So, the result is correct.