Find the integral int(1/(3969+49x^2))dx

 Exercise

\int\frac{1}{3969+49x^2}\:dx

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Rewrite the expression $\frac{1}{3969+49x^2}$ inside the integral in factored form

\int\frac{1}{49\left(81+x^2\right)}dx

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Take the constant $\frac{1}{49}$ out of the integral

\frac{1}{49}\int\frac{1}{81+x^2}dx

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Solve the integral by applying the formula $\displaystyle\int\frac{x'}{x^2+a^2}dx=\frac{1}{a}\arctan\left(\frac{x}{a}\right)$

\frac{1}{49}\cdot \frac{1}{9}\arctan\left(\frac{x}{9}\right)

 

Multiplying fractions $\frac{1}{49} \times \frac{1}{9}$

\frac{1}{441}\arctan\left(\frac{x}{9}\right)

 

As the integral that we are solving is an indefinite integral, when we finish integrating we must add the constant of integration $C$

\frac{1}{441}\arctan\left(\frac{x}{9}\right)+C_0

\frac{1}{441}\arctan\left(\frac{x}{9}\right)+C_0

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