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5ezpieces 5ezpieces
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12 years ago
Question 1-
Evaluate
|x^2  square root(x+3)  dx.


Question 2-
Using all the necessary tools of calculus, sketch the graph and solve f(x)=x^4/4 - 4/3x^3 + 2x^2 - 1.
Identifying all relative extrema and all inflection points. In addition cite intervals of monotonicity and concavity.
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12 years ago
Integral ( x^2 sqrt(x + 3) dx )

To solve this, use substitution.

Let u = sqrt(x + 3).  Square both sides,

u^2 = x + 3.  Bring the +3 to the left hand side,

u^2 - 3 = x.  Differentiate both sides.

2u du = dx

As a result of our substitution, we get

Integral ( (u^2 - 3)^2 u (2u) du )

Factor out the 2,

2 Integral ( u^2 - 3)^2 (u^2) du )

Square the binomial.

2 * Integral ( (u^4 - 6u^2 + 9)u^2 du )

Distribute the u^2.

2 * Integral ( (u^6 - 6u^4 + 9u^2)  du )

Integrate using the reverse power rule.

2 [ (1/7)u^7 - 6(1/5)u^5 + 9(1/3)u^3 ] + C

2 [ (1/7)u^7 - (6/5)u^5 + 3u^3 ] + C

Distribute the 2,

(2/7)u^7 - (12/5)u^5 + 6u^3 + C

And now, resubstitute u = sqrt(x + 3).

(2/7) [sqrt(x + 3)]^7 - (12/5)[sqrt(x + 3)]^5 + 6[sqrt(x + 3)]^3 + C

Using the fact that sqrt is the same as ^(1/2), and multiplying exponents and what not, the above should simplify to

(2/7) (x + 3)^(7/2) - (12/5) (x + 3)^(5/2) + 6(x + 3)^(3/2) + C
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