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High School
Calculus and Analysis
Calculus 2
Applications of integrals

Irrerbthist6n

Answered question

2021-12-14

Find formulas for the functions represented by the integrals.

\int_{1}^{x^{2}} t d t

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}} t d t

Step 2:Formula used

\int x^{n} d x = \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}} t d t

= \frac{t^{2}}{2} ∣_{1}^{x^{2}}

= \frac{1}{2} ({(x^{2})}^{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}} t d t

Evaluate the indefinite integral

\int t d t

Evaluate the integral

\frac{t^{2}}{2}

Return the limits

\frac{t^{2}}{2} ∣_{1}^{x^{2}}

Calculate the expression

\frac{{(x^{2})}^{2}}{2} - \frac{1^{2}}{2}

Simplify
Solution

\frac{x^{4} - 1}{2}

nick1337

Expert2021-12-28Added 777 answers

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

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