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Ksju [112]
1 year ago
11

Integrate this two questions ​

Mathematics
1 answer:
tankabanditka [31]1 year ago
5 0

Simplify the integrands by polynomial division.

\dfrac{t^2}{1 - 3t} = -\dfrac19 \left(3t + 1 - \dfrac1{1 - 3t}\right)

\dfrac t{1 + 4t} = \dfrac14 \left(1 - \dfrac1{1 + 4t}\right)

Now computing the integrals is trivial.

5.

\displaystyle \int \frac{t^2}{1 - 3t} \, dt = -\frac19 \int \left(3t + 1 - \frac1{1-3t}\right) \, dt \\\\ = -\frac19 \left(\frac32 t^2 + t + \frac13 \ln|1 - 3t|\right) + C \\\\ = \boxed{-\frac{t^2}6 - \frac t9 - \frac{\ln|1-3t|}{27} + C}

where we use the power rule,

\displaystyle \int x^n \, dx = \frac{x^{n+1}}{n+1} + C ~~~~ (n\neq-1)

and a substitution to integrate the last term,

\displaystyle \int \frac{dt}{1-3t} = -\frac13 \int \frac{du}u \\\\ = -\frac13 \ln|u| + C \\\\ = -\frac13 \ln|1-3t| + C ~~~ (u=1-3t \text{ and } du = -3\,dt)

8.

\displaystyle \int \frac t{1+4t} \, dt = \frac14 \int \left(1 - \frac1{1+4t}\right) \, dt \\\\ = \frac14 \left(t - \frac14 \ln|1 + 4t|\right) + C \\\\ = \boxed{\frac t4 - \frac{\ln|1+4t|}{16} + C}

using the same approach as above.

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Carmen saved 592 pennies. Her sister saved 128 pennies. Together, they put 250 pennies in wrappers and took them to the bank. Wh
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500 pennies

Step-by-step explanation:

Given that

Carmen saved 592 pennies.

Her sister saved 128 pennies.

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We need to find out the total number of pennies Carmen and her sister have left

So,

= 592 pennies + 128 pennies  - 250 pennies

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7 0
2 years ago
Brian spends 3 5 of his wages on rent and 1 3 on food. If he makes £735 per week, how much money does he have left?
Margarita [4]

Answer :

Brian has $49 left.

Step by step explanation :

Step 1 : Find how much money Brian spends on rent and food.

First question we should be asking is what is 3/5 of 735?

Let's find that.

\frac{3}{5} * \frac{735}{1}

= \frac{(3)(735)}{(5)(1)}

= \frac{2205}{5}

= 441

We know Brian spends $441 on rent. Now we find how much he spends on food.

\frac{1}{3} * \frac{735}{1}

= \frac{(1)(735)}{(3)(1)}

= \frac{735}{3}

= 245

Brian spends $245 on food.

Step 2 : Add what he spends on rent and food together.

$441 + $245 = $686

Step 3 : To find out how much money Brian has left, we subtract $686 from $735.

$735 - $686 = $49

Conclusion : Brian has $49 left.

Hope this helps, please mark brainliest if possible. Have a nice day.

3 0
2 years ago
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