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January 27, 2015

January 27, 2015

Posted by **CMM** on Thursday, January 27, 2011 at 11:15pm.

S= integral symbol

S x^3*sqrt(x^2 + 1) dx

u = x^2+1

x^2= u - 1

du = 2xdx

du/2 = xdx

1/2 S x^2 * x sqrt (x^2 + 1) du

1/2 S (u - 1) * sqrt (u) du multiply square root of u and (u - 1)

FOIL (u - 1) * sqrt (u)

=((u)^3/2) - ((u)^1/2)

= 1/2 S (((u)^3/2) - ((u)^1/2)) du Now take the derivative and keep the 1/2 on the side.

1/2 ((u)^5/2)/(5/2) - ((u)^3/2)/(3/2) now distribute the 1/2 and bring the 5/2 and 3/2 to the top

1/2 * 2/5 * (u)^5/2 - 1/2 * 2/3 * (u)^ 3/2

cut all the 2 and substitute x^2+1 in the place of u.

so the final answer is

1/5 (x^2 - 1)^5/2 - 1/3(x^2 - 1)^3/2

- Calc. -
**helper**, Thursday, January 27, 2011 at 11:59pm| x^3*sqrt(x^2 + 1) dx

| = integral sign

| x^2 (sqrt(x^2 + 1)) x dx

u = x^2

du = 2x dx

1/2 du = x dx

1/2 | u (sqrt(u + 1)) du

w = u + 1

dw = du

u = w - 1

1/2 | (w - 1) (sqrt(w)) dw

1/2 | (w - 1) w^1/2 dw

1/2 | w^3/2 dw - 1/2 | w^1/2 dw

1/2 (2/5 w^5/2) - 1/2 ( 2/3 w^3/2) + C

2/10 w^5/2 - 2/6 w^3/2 + C

1/5 w^5/2 - 1/3 w^3/2 + C

w = u + 1

1/5 (u + 1)^5/2 - 1/3 (u + 1)^3/2 + C

u = x^2

1/5 (x^2 + 1)^5/2 - 1/3 (x^2 + 1)^3/2 +

C

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