We are given the following indefinite integral.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(4-\frac{1}{t^5}{)}^2(\frac{1}{t^6})dt$

Integration.

Let $u=4-\frac{1}{t^5}.$

Find $du.$

$du=(,)dt$

Find the indefinite integral and check the result by differentiation.

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(4-\frac{1}{t^5}{)}^2(\frac{1}{t^6})dt$

Step $1$

We are given the following indefinite integral

${\displaystyle \underset{\phantom{}}{\overset{\phantom{}}{}}}(4-\frac{1}{t^2}{)}^2(\frac{1}{t^6})dt$

The integrand is a product of two quantities, with one raised to a power. If the second quantity can be written as the derivative of the quantity raised to a power, then a substitution will allow us to use the General Power Rule for Integration

Let $U=4-\frac{1}{5^5}$

Find du$.$

${?}^{du}=|)$ dt