Answered: "Solve the integral. ∫19t−12dt"

grislingatb

Answered question

2021-11-23

Solve the integral.

\int_{1}^{9} t^{- \frac{1}{2}} d t

Drood1980

Beginner2021-11-24Added 16 answers

Step 1
To find:
The definite integral of

\int_{1}^{9} t^{- \frac{1}{2}} d t

.
Formula used:
Power rule of integration:

{(x^{n})}^{'} = n x^{n - 1}

Calculation:
The definite integral of

\int_{1}^{9} t^{- \frac{1}{2}} d t

can be obtained as,

\int_{1}^{9} t^{- \frac{1}{2}} d t = {[\frac{t^{- \frac{1}{2} + 1}}{- \frac{1}{2} + 1}]}_{1}^{9}

= 2 {[\sqrt{t}]}_{1}^{9}

= 2 [\sqrt{9} - \sqrt{1}]

=4
Step 2
Thus, the integral of

\int_{1}^{9} t^{- \frac{1}{2}} d t

is 4.

Kathleen Ashton

Beginner2021-11-25Added 15 answers

Step 1: Use Negative Power Rule:

x^{- a} = \frac{1}{x^{a}}

\int_{1}^{9} \frac{1}{\sqrt{t}} d t

Step 2: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:

\int_{a}^{b} f (x) d x = F (x) ∣_{a}^{b} = F (b) - F (a)

Step 3: In this case,

f (t) = \frac{1}{\sqrt{t}}

. Find its integral.

2 \sqrt{t} ∣_{1}^{9}

Step 4: Since

F (t) ∣_{a}^{b} = F (b) - F (a)

, expand the above into F(9)−F(1):

2 \sqrt{9} - 2 \sqrt{1}

Step 5: Simplify.
4

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