# int x^2sqrt(2+9x^2) dx Use integration tables to find the indefinite integral. Recall that indefinite integral follows int f(x) dx = F(x) +C

where:

f(x) as the integrand function

F(x) as the antiderivative of f(x)

C as the constant of integration.

The given integral problem: int x^2sqrt(2+9x^2) dx  resembles one of the formulas from the integration table. We follow the integral formula for...

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Recall that indefinite integral follows int f(x) dx = F(x) +C

where:

f(x) as the integrand function

F(x) as the antiderivative of f(x)

C as the constant of integration.

The given integral problem: int x^2sqrt(2+9x^2) dx  resembles one of the formulas from the integration table. We follow the integral formula for function with roots as:

int u^2sqrt(a^2+u^2)du = u/8(a^2+2u^2)sqrt(a^2+u^2) -a^4/8ln|u+sqrt(a^2+u^2)| +C .

For easier comparison, we apply u-substitution by letting:

u^2 = 9x^2 or (3x)^2

then u = 3x  or x=u/3 .

For the derivative of u, we get: du = 3 dx or (du)/3 = dx .

Note: The corresponding value of a^2=2

then a =sqrt(2) and   a^4 = (a^2)^2 =2^2 =4

Plug-in the values of u = 3x , x=u/3 and (du)/3 = dx , we get:

int x^2sqrt(2+9x^2)dx=int (u/3)^2sqrt(2+u^2)* (du)/3

=int u^2/9*sqrt(2+u^2)* (du)/3

=int u^2/27sqrt(2+u^2)du

Apply the basic integration property: int c*f(x) dx = c int f(x) dx .

int u^2/27*sqrt(2+u^2)*du =1/27int u^2sqrt(2+u^2)du

Apply the aforementioned integral formula with a^2 =2 , we get:

1/27 int u^2sqrt(2+u^2)du=1/27*[u/8(2+2u^2)sqrt(2+u^2) -4/8ln|u+sqrt(2+u^2)|]+C

= u/216(2+2u^2)sqrt(2+u^2) -1/2ln|u+sqrt(2+u^2)|+C

Plug-in u =3x on  u/216(2+2u^2)sqrt(2+u^2) -1/2ln|u+sqrt(2+u^2)|+C , we get the indefinite integral as:

int x^2sqrt(2+9x^2) dx=(3x)/216(2+2(3x)^2)sqrt(2+(3x)^2) -1/2ln|3x+sqrt(2+(3x)^2)|+C

= x/72(2+18x^2)sqrt(2+9x^2) -1/2ln|3x+sqrt(2+9x^2)|+C

= ((2x+18x^3)sqrt(2+9x^2))/72 -(ln|3x+sqrt(2+9x^2)|)/2+C

or (xsqrt(2+9x^2))/36+(x^3sqrt(2+9x^2))/4 -(ln|3x+sqrt(2+9x^2)|)/2+C

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