What is the integral of 1/(x^2sqrt(7-x^2)) ?

The integral of 1/(x^2sqrt(7-x^2)) is -(sqrt(7-x^2))/(7x)+C

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integral of 1/(x^2sqrt(7-x^2))

\int \frac{1}{x ^{2} 7 - x ^{2}} d x

- \frac{7 - x ^{2}}{7 x} + C

Solution steps

\int \frac{1}{x ^{2} 7 - x ^{2}} d x

Apply Trigonometric Substitution

: \int \frac{cot ( u )}{7 sin ( u ) cos ( u )} d u

= \int \frac{cot ( u )}{7 sin ( u ) cos ( u )} d u

Take the constant out:

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

= \frac{1}{7} \cdot \int \frac{cot ( u )}{sin ( u ) cos ( u )} d u

Simplify

\frac{cot ( u )}{sin ( u ) cos ( u )} : csc^{2} (u)

= \frac{1}{7} \cdot \int csc^{2} (u) d u

Use the common integral:

\int csc^{2} (u) d u = - cot (u)

= \frac{1}{7} (- cot (u))

Substitute back

u = arcsin (\frac{1}{7} x)

= \frac{1}{7} (- cot (arcsin (\frac{1}{7} x)))

Simplify

\frac{1}{7} (- cot (arcsin (\frac{1}{7} x))) : - \frac{7 - x ^{2}}{7 x}

= - \frac{7 - x ^{2}}{7 x}

Add a constant to the solution

= - \frac{7 - x ^{2}}{7 x} + C

\frac{\partial}{\partial t} (\frac{( s - r )}{t})

x \to 10 lim (\frac{( 3 x ^{2} - 30 x )}{x ^{2} - 100})

\int \frac{5 x + 1}{2 x ^{2} - x - 1} d x

2 x e^{- x}

\frac{d}{d x} (\frac{5}{12} x^{\frac{6}{5}} - \frac{5}{8} x^{\frac{4}{5}})

What is the integral of 1/(x^2sqrt(7-x^2)) ?
The integral of 1/(x^2sqrt(7-x^2)) is -(sqrt(7-x^2))/(7x)+C