`int(dx)/(5x^2+7)=`

Options

A. `(1)/(sqrt(35))tan^(-1)(sqrt(5x)/(sqrt7))+c`
B. `(1)/(5sqrt(7))tan^(-1)(sqrt(5x)/(sqrt7))+c`
C. `(sqrt(5)/(sqrt(7)))tan^(-1)(sqrt(5x)/(sqrt7))+c`
D. `(5)/(sqrt(7))tan^(-1)(sqrt(5x)/(sqrt7))+c`

Answer

To solve the integral \( \int \frac{dx}{5x^2 + 7} \), we can follow these steps:

### Step 1: Rewrite the Integral
We start with the integral:
\[ I = \int \frac{dx}{5x^2 + 7} \]

### Step 2: Factor Out the Constant
We can factor out the constant from the denominator:
\[ I = \int \frac{dx}{5\left(x^2 + \frac{7}{5}\right)} = \frac{1}{5} \int \frac{dx}{x^2 + \frac{7}{5}} \]

### Step 3: Identify the Standard Form
Now, we recognize that the integral \( \int \frac{dx}{x^2 + a^2} \) has a standard result:
\[ \int \frac{dx}{x^2 + a^2} = \frac{1}{a} \tan^{-1} \left(\frac{x}{a}\right) + C \]
In our case, \( a^2 = \frac{7}{5} \), so \( a = \sqrt{\frac{7}{5}} \).

### Step 4: Substitute into the Standard Form
Using the standard form, we substitute \( a \):
\[ I = \frac{1}{5} \cdot \frac{1}{\sqrt{\frac{7}{5}}} \tan^{-1} \left(\frac{x}{\sqrt{\frac{7}{5}}}\right) + C \]

### Step 5: Simplify the Expression
Now simplify \( \frac{1}{\sqrt{\frac{7}{5}}} \):
\[ \frac{1}{\sqrt{\frac{7}{5}}} = \frac{\sqrt{5}}{\sqrt{7}} \]
Thus, we have:
\[ I = \frac{1}{5} \cdot \frac{\sqrt{5}}{\sqrt{7}} \tan^{-1} \left(\frac{x \sqrt{5}}{\sqrt{7}}\right) + C \]

### Step 6: Final Result
The final result for the integral is:
\[ I = \frac{\sqrt{5}}{5\sqrt{7}} \tan^{-1} \left(\frac{x \sqrt{5}}{\sqrt{7}}\right) + C \]