Integrate int(dx)/(7x^(2)+8)

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Identify Integral: Identify the integral that needs to be solved. We need to find the indefinite integral of the function 1/(7x^2+8) with respect to x.Recognize Form: Recognize the form of the integral. The integral is of the form ∫dx/(ax^2+b), which suggests a trigonometric substitution or a direct comparison to the standard integral form ∫dx/(x^2+a^2), which results in (1/a)arctan(x/a) + C.Perform Substitution: Perform a substitution to match the standard form. Let's compare 7x^2+8 to the standard form x^2+a^2. We can rewrite the integral as ∫dx/(7(x^2+(8/7))).Factor Out Constant: Factor out the constant from the denominator. We can factor out the 7 from the denominator to match the standard form. The integral becomes (1/7)∫dx/(x^2+(8/7)).Complete Square: Complete the square in the denominator. To complete the square, we need to have a perfect square in the denominator. We already have that since (8/7) is a constant term. So, we can write the integral as (1/7)∫dx/(x^2+(√(8/7))^2).Apply Standard Form: Apply the standard integral form. Now we can apply the standard integral form ∫dx/(x^2+a^2) = (1/a)arctan(x/a) + C. Here, a = √(8/7), so the integral becomes (1/√(8/7))arctan(x/√(8/7)) + C.Simplify Expression: Simplify the expression. Simplify the expression to get the final answer. We have (1/√(8/7))arctan(x/√(8/7)) + C = (√(7/8))arctan(x√(7/8)) + C.

Integrate $\int \frac{d x}{7 x^{2}+8}$ =

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Integrate ∫dx7x2+8 \int \frac{d x}{7 x^{2}+8} ∫7x2+8dx​=

Integrate $\int \frac{d x}{7 x^{2}+8}$ =