Find formulas for the functions represented by the integrals. \int_{1}^{x^{2}}t dt

Step 1: To determine
Find formula for the function represented by the given integral:
${\displaystyle {\int }_1^{x^2}tdt}$
Step 2:Formula used
${\displaystyle \int x^{n}dx=\frac{x^{n+1}}{n+1}+C}$ where C is the constant of integration
Step 3:Solution
Consider the given integral:
${\displaystyle {\int }_1^{x^2}tdt}$
${\displaystyle =\frac{t^2}{2}{\mid }_1^{x^2}}$
${\displaystyle =\frac{1}{2}({\left(x^2\right)}^2-1^2)}$
${\displaystyle =\frac{1}{2}(x^4-1)}$
Hence, the function represented by the given integral is ${\displaystyle \frac{1}{2}(x^4-1)}$
Step 4:Conclusion
Hence, the function represented by the given integral is ${\displaystyle \frac{1}{2}(x^4-1)}$