Explain how you can tell without calculating whether the sum of a positive number and a negative number will be positive, negati
2 answers:
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The correct answers are:
(1) If EXACT number is required, then 7.5.
(2) If multiple of 8 is required then the answer will be 7 times with a remainder of 4.
Explanation:
First you need to express it in the form of equation to make things simpler as follows:
8 * y = 60
We need to find y; to do so, divide 60 by 8:
y =
y = 7.5
Now if you want to find the whole number, then the answer will be 7 times, with 4 left over.
Derivative of the denominator:
Hmm our numerator is 2x+7. Ok this let's us know that a simple u-substitution is NOT going to work. But let's apply some clever Algebra to the numerator splitting it up into two separate fractions. Split the +7 into +2 and +5.
and then split the fraction,
Based on our previous test, we know that a simple substitution will work for the first integral:
So the first integral changes,
integrating to a log,
Other one is a little tricky. We'll need to complete the square on the denominator. After that it will look very similar to our arctangent integral so perhaps we can just match it up to the identity.
So we have this going on,
Let's factor the 5 out of the intergral,
and the 4 from the denominator,
Bringing all that stuff together as a single square,
Making the substitution:
giving us,
simplying a lil bit,
and hopefully from this point you recognize your arctangent integral,
undo your substitution as a final step,
and include a constant of integration,
Hope that helps!
Lemme know if any steps were too confusing.
