Questions: ∫ 5x √(5x^2-5) dx

Transcript text: $\int 5 x \sqrt{5 x^{2}-5} d x$

Solution

Solution Steps

To solve the integral $\int 5x \sqrt{5x^2 - 5} \, dx$, we can use a substitution method. Let's set $u = 5x^2 - 5$, which simplifies the expression under the square root. Then, compute $du$ in terms of $dx$ and substitute back into the integral to solve it in terms of $u$.

Step 1: Identify the Integral and Use Substitution

To solve the integral $\int 5x \sqrt{5x^2 - 5} \, dx$, we start by using a substitution method. Let $u = 5x^2 - 5$. Then, the derivative $du$ with respect to $x$ is $du = 10x \, dx$, or $dx = \frac{du}{10x}$.

Step 2: Substitute and Simplify

Substitute $u$ and $dx$ into the integral: \[ \int 5x \sqrt{u} \cdot \frac{du}{10x} = \frac{1}{2} \int \sqrt{u} \, du \]

Step 3: Integrate with Respect to $u$

The integral $\frac{1}{2} \int \sqrt{u} \, du$ can be solved as: \[ \frac{1}{2} \cdot \frac{2}{3} u^{3/2} = \frac{1}{3} u^{3/2} \]

Step 4: Substitute Back in Terms of $x$

Substitute back $u = 5x^2 - 5$ into the expression: \[ \frac{1}{3} (5x^2 - 5)^{3/2} \]

Step 5: Simplify the Expression

Simplify the expression to match the output: \[ \frac{5\sqrt{5}}{3} x^2 \sqrt{x^2 - 1} - \frac{5\sqrt{5}}{3} \sqrt{x^2 - 1} \]

Final Answer

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

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