Questions: ∫ 28/(196x^2+1)^2 dx

Solution Steps

To solve the integral \(\int \frac{28}{\left(196 x^{2}+1\right)^{2}} d x\), we can use a trigonometric substitution or recognize it as a standard form of an integral. The expression inside the integral resembles the derivative of the arctangent function. We can use a substitution to simplify the integral and then apply the standard integral formula for \(\int \frac{1}{(a^2 + x^2)^2} dx\).

The given integral is \(\int \frac{28}{(196x^2 + 1)^2} \, dx\). This integral can be solved by recognizing it as a form that involves the derivative of the arctangent function and a rational function.

The expression \(\frac{28}{(196x^2 + 1)^2}\) can be simplified by factoring out constants and recognizing the standard integral form. We can rewrite the integral as: \[ \int \frac{28}{(196x^2 + 1)^2} \, dx = \int \frac{28}{(14^2x^2 + 1)^2} \, dx \]

The integral \(\int \frac{1}{(a^2 + x^2)^2} \, dx\) is a standard form, and its solution involves the arctangent function. Using this, we can find the antiderivative: \[ \int \frac{28}{(196x^2 + 1)^2} \, dx = \frac{28x}{392x^2 + 2} + \arctan(14x) + C \]

Final Answer