# Evaluate : ∫x 2√5x − 3dx

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To evaluate the integral $$\int x \sqrt{5x - 3} \, dx$$, we can use u-substitution.

Let $$u = 5x - 3$$, then $$du/dx = 5$$ or $$dx = du/5$$.

Now, substitute $$u$$ and $$dx$$ into the integral:

$\int x \sqrt{5x - 3} \, dx = \int x \sqrt{u} \, \frac{du}{5}$

Now, we can pull the constant (1/5) out of the integral:

$\frac{1}{5} \int x \sqrt{u} \, du$

Next, we can further simplify by expressing $$x$$ in terms of $$u$$ using $$u = 5x - 3$$, which gives $$x = \frac{u + 3}{5}$$.

Substitute this into the integral:

$\frac{1}{5} \int \frac{u + 3}{5} \sqrt{u} \, du$

Now, distribute the $$\frac{1}{5}$$ into the expression:

$\frac{1}{25} \int (u + 3) \sqrt{u} \, du$

Expand the expression:

$\frac{1}{25} \int (u^{3/2} + 3u^{1/2}) \, du$

Now, integrate each term separately:

$\frac{1}{25} \left( \frac{2}{5}u^{5/2} + 2u^{3/2} \right) + C$

Finally, substitute back $$u = 5x - 3$$ to get the final result:

$\frac{1}{25} \left( \frac{2}{5}(5x - 3)^{5/2} + 2(5x - 3)^{3/2} \right) + C$

where $$C$$ is the constant of integration.