Question

In Exercises \(25-46,\) use substitution to evaluate the integral. $$\int \frac{40 d x}{x^{2}+25}$$

Short answer

The solution to \(\int \frac{40 dx}{x^{2}+25}\) is \(\frac{8x}{5} + C\).

Step-by-step solution

Substitution

Let's make a substitution. Let \(x = 5\tan(\theta)\). Now the next step is to differentiate x with respect to \(\theta\). This gets \(\frac{dx}{d\theta} = 5\sec^2(\theta)\), or \(dx = 5\sec^2(\theta)d\theta\).

Transform the Integral

Substituting these into the integral, we get \(\int \frac{40 dx}{x^2 + 25} = \int \frac{40*5\sec^2(\theta) d\theta}{(5\tan(\theta))^2 + 25}\). Now simplify this to \(\int 8\sec^2(\theta) d\theta\).

Integration

Now we need to integrate this. Recognizing \(\sec^2(\theta)\) as the derivative of \(\tan(\theta)\), the integral of 8sec²(\(\theta\)) \(d\theta\) is 8tan(\(\theta\)) + C.

Substitute back

Substituting the original variable back, we substitute \(x = 5\tan(\theta)\) to get \(\frac{8x}{5} + C\).