Questions: Compute the given integral. [ int frac1x^2+49 d x= ] +C

$Compute the given integral. [ int frac1x^2+49 d x= ] +C$

Transcript text: Compute the given integral. \[ \int \frac{1}{x^{2}+49} d x= \] $\square$ \[ +C \] Submit Question

Solution

Solution Steps

To solve the integral $\int \frac{1}{x^{2}+49} \, dx$, we can recognize that it resembles the standard integral form $\int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$. Here, $a = 7$.

Step 1: Recognize the Integral Form

The given integral is: \[ \int \frac{1}{x^{2}+49} \, dx \] We recognize that this integral is of the form: \[ \int \frac{1}{a^2 + x^2} \, dx = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C \] where $a = 7$.

Step 2: Apply the Integral Formula

Using the formula, we substitute $a = 7$: \[ \int \frac{1}{x^{2}+49} \, dx = \frac{1}{7} \arctan\left(\frac{x}{7}\right) + C \]