# int(x dx )/ (2x^2 +3)^5 Evaluate the integral

Neethu | Certified Educator

We have to evaluate the integral \int \frac{xdx}{(2x^2+3)^5}

Let us use the substitution:

2x^2+3=t

Differentiating the above, we get,

4xdx=dt

xdx=\frac{dt}{4}

Therefore, we can write the integral as,

\int \frac{xdx}{2x^2+3}=\frac{1}{4}\int \frac{dt}{t^5}

=\frac{1}{4}\int t^{-5}dt 

=\frac{1}{4}.\frac{t^{-5+1}}{-5+1}+C where C is a constant.

=\frac{-1}{16t^4}+C

=\mathbf{\frac{-1}{16(2x^2+3)^4}} +C

mathewww | Student

To find int (xdx)/(2x^2+3)^5

Let u=2x^2+3 , then du=4x dx rArr 1/4du=xdx . The substitution gives:

int (xdx)/(2x^2+3)^5

=int 1/(4u^5) du

=int 1/4*u^-5 du

Applying the formula int x^n dx = x^(n+1)/(n+1)+C we get:

1/4*u^(-5+1)/(-5+1)+C

=1/4*u^-4/-4+C

=-1/(16u^4)+C

Replacing u with 2x^2+3 , we get the required solution as:

-1/(16(2x^2+3)^4)+C.

steamgirl | Student

For this problem you want to use U-substitution. Set u = to 2x^2 + 3, therefore du = 4x dx, or du/4 = x dx.

(1/4) the integral of u^-5 du.

Which with the power rule, would mean you add one to the exponent and divide by the resulting number. This gives you:

(-1/16)(1/(2x^2 + 3)^4)

simplelucker | Student

To find

Let  , then  . The substitution gives:

Applying the formula  we get:

Replacing  with  , we get the required solution as:

kailash | Student

int(xdx)/(2x^2+3)^5

We can asnwer this integration by method of substitution.

t=2x^2+3

dt=4xdx

(1/4)dt=xdx

Thus,

int(xdx)/(2x^2+3)^5=int(dt/4)(1/t^5)

=(1/4)intdt/t^5

=(1/4)int t^(-5)dt

=(1/4)t^(-5+1)/(-5+1)+c

=(-1/16)t^(-4)+c

=-1/16(2x^2+3)^(-4)+c

where c is integrating constant.