integrate this without using u substitution . integrate 1/(25x^2+1)

To integrate \( \int \frac{1}{25x^2 + 1} \, dx \) without using \( u \)-substitution, let's proceed by recognizing a form that resembles the derivative of an inverse trigonometric function.

The integral has the form \( \int \frac{1}{a^2 x^2 + 1} \, dx \), which is similar to the form for the inverse tangent function. Recall that:

\[
\int \frac{1}{a^2 x^2 + 1} \, dx = \frac{1}{a} \arctan(ax) + C
\]

In our case, \( a = 5 \) (since \( 25x^2 = (5x)^2 \)). Applying this directly, we get:

\[
\int \frac{1}{25x^2 + 1} \, dx = \frac{1}{5} \arctan(5x) + C
\]

where \( C \) is the constant of integration.

Would you like a more detailed explanation or have any other questions?

Here are some related questions you might find interesting:
1. How do we determine when an integral matches the form of inverse trigonometric functions?
2. How can we handle integrals that do not fit standard forms?
3. What are other techniques for integration without substitution?
4. How does the derivative of an inverse tangent relate to this integral?
5. What is the general formula for integrating functions of the form \( \frac{1}{a^2x^2 + b} \)?

**Tip:** Familiarizing yourself with standard integral forms can save time by identifying patterns in integrands directly.

Learn how to integrate the function 1/(25x^2 + 1) without using u-substitution. This problem uses the formula for the inverse tangent function. It is a standard calculus problem for students in Grades 11-12.

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