Obtain an expression for the $n$th term of an asymptotic expansion, valid as $\lambda \rightarrow \infty$, for the integral

$$I(\lambda)=\int_{0}^{1} t^{2 \alpha} e^{-\lambda\left(t^{2}+t^{3}\right)} d t \quad(\alpha>-1 / 2) .$$

Estimate the value of $n$ for the term of least magnitude.

Obtain the first two terms of an asymptotic expansion, valid as $\lambda \rightarrow \infty$, for the integral

$$J(\lambda)=\int_{0}^{1} t^{2 \alpha} e^{-\lambda\left(t^{2}-t^{3}\right)} d t \quad(-1 / 2<\alpha<0)$$

[Hint:

$$\left.\Gamma(z)=\int_{0}^{\infty} t^{z-1} e^{-t} d t .\right]$$

[Stirling's formula may be quoted.]