Questions: ∫ x √(25x-8) dx

Solution Steps

Let \( u = 25x - 8 \). Then, we differentiate to find \( du = 25 \, dx \) or \( dx = \frac{du}{25} \). We also express \( x \) in terms of \( u \): \( x = \frac{u + 8}{25} \).

Substituting \( u \) and \( dx \) into the integral, we have: \[ \int x \sqrt{25x - 8} \, dx = \int \left(\frac{u + 8}{25}\right) \sqrt{u} \cdot \frac{du}{25} \] This simplifies to: \[ \frac{1}{625} \int (u + 8) \sqrt{u} \, du \]

Now, we can split the integral: \[ \frac{1}{625} \left( \int u^{3/2} \, du + 8 \int u^{1/2} \, du \right) \] Calculating these integrals gives: \[ \int u^{3/2} \, du = \frac{2}{5} u^{5/2} \quad \text{and} \quad \int u^{1/2} \, du = \frac{2}{3} u^{3/2} \] Thus, we have: \[ \frac{1}{625} \left( \frac{2}{5} u^{5/2} + \frac{16}{3} u^{3/2} \right) \]

Finally, substituting back \( u = 25x - 8 \) into the expression yields: \[ \frac{1}{625} \left( \frac{2}{5} (25x - 8)^{5/2} + \frac{16}{3} (25x - 8)^{3/2} \right) + C \] This represents the evaluated integral.

Final Answer