What is the integral of 1/(t^2sqrt(t^2-a^2)) ?

The integral of 1/(t^2sqrt(t^2-a^2)) is (sqrt(t^2-a^2))/(a^2t)+C

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integral of 1/(t^2sqrt(t^2-a^2))

\int \frac{1}{t ^{2} t ^{2} - a ^{2}} d t

\frac{t ^{2} - a ^{2}}{a ^{2} t} + C

Solution steps

\int \frac{1}{t ^{2} t ^{2} - a ^{2}} d t

Apply Trigonometric Substitution

: \int \frac{1}{a ^{2} sec ( u )} d u

= \int \frac{1}{a ^{2} sec ( u )} d u

Take the constant out:

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

= \frac{1}{a ^{2}} \cdot \int \frac{1}{sec ( u )} d u

Rewrite using trig identities

= \frac{1}{a ^{2}} \cdot \int cos (u) d u

Use the common integral:

\int cos (u) d u = sin (u)

= \frac{1}{a ^{2}} sin (u)

Substitute back

u = arcsec (\frac{1}{a} t)

= \frac{1}{a ^{2}} sin (arcsec (\frac{1}{a} t))

Simplify

\frac{1}{a ^{2}} sin (arcsec (\frac{1}{a} t)) : \frac{t ^{2} - a ^{2}}{a ^{2} t}

= \frac{t ^{2} - a ^{2}}{a ^{2} t}

Add a constant to the solution

= \frac{t ^{2} - a ^{2}}{a ^{2} t} + C

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