What is the integral of 1/(x^2sqrt(4x^2+49)) ?

The integral of 1/(x^2sqrt(4x^2+49)) is -(sqrt(49+4x^2))/(49x)+C

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integral of 1/(x^2sqrt(4x^2+49))

\int \frac{1}{x ^{2} 4 x ^{2} + 49} d x

- \frac{49 + 4 x ^{2}}{49 x} + C

Solution steps

\int \frac{1}{x ^{2} 4 x ^{2} + 49} d x

Apply Trigonometric Substitution

: \int \frac{2 sec ( u )}{49 tan ^{2} ( u )} d u

= \int \frac{2 sec ( u )}{49 tan ^{2} ( u )} d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= \frac{2}{49} \cdot \int \frac{sec ( u )}{tan ^{2} ( u )} d u

Rewrite using trig identities

= \frac{2}{49} \cdot \int \frac{cos ( u )}{sin ^{2} ( u )} d u

Apply u-substitution

: \int \frac{1}{v ^{2}} d v

= \frac{2}{49} \cdot \int \frac{1}{v ^{2}} d v

Apply the Power Rule

: - \frac{1}{v}

= \frac{2}{49} (- \frac{1}{v})

Substitute back

= \frac{2}{49} (- \frac{1}{sin ( arctan ( \frac{2}{7} x ) )})

Simplify

\frac{2}{49} (- \frac{1}{sin ( arctan ( \frac{2}{7} x ) )}) : - \frac{49 + 4 x ^{2}}{49 x}

= - \frac{49 + 4 x ^{2}}{49 x}

Add a constant to the solution

= - \frac{49 + 4 x ^{2}}{49 x} + C

\int \frac{- x ^{2} - 1}{x ^{2} - 1} d x

9 e^{x} (cos (x) - sin (x))

n = 0 \sum \infty (- x)^{n}

x \to 0 + lim (\frac{x}{ln ( \frac{1}{x} )})

\frac{\partial}{\partial x} (\frac{y}{2 x})

What is the integral of 1/(x^2sqrt(4x^2+49)) ?
The integral of 1/(x^2sqrt(4x^2+49)) is -(sqrt(49+4x^2))/(49x)+C