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

Irrerbthist6n

Answered question

2021-12-14

Find formulas for the functions represented by the integrals.
${\int}_{1}^{{x}^{2}}tdt$

Answer & Explanation

Ronnie Schechter

Beginner2021-12-15Added 27 answers

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

Shannon Hodgkinson

Beginner2021-12-16Added 34 answers

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

nick1337

Expert2021-12-28Added 573 answers

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