Find formulas for the functions represented by the integrals. ∫1x2tdt

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 ${\displaystyle \frac{1}{2}(x^4-1)}$

${\displaystyle {\int }_1^{x^2}tdt}$
Evaluate the indefinite integral
${\displaystyle \int tdt}$
Evaluate the integral
${\displaystyle \frac{t^2}{2}}$
Return the limits
${\displaystyle \frac{t^2}{2}{\mid }_1^{x^2}}$
Calculate the expression
${\displaystyle \frac{{\left(x^2\right)}^2}{2}-\frac{1^2}{2}}$
Simplify
Solution
${\displaystyle \frac{x^4-1}{2}}$

It is required to calculate:
$\int tdt$
Integral of a power function:
$\int t^{n}dt=\frac{t^{n+1}}{n+1}atn=1:$
$=\frac{t^2}{2}$
Problem solved:
$\int tdt$
$=\frac{t^2}{2}+C$