What is the integral from 1 to tan(5x) of sqrt(1-t) ?

The integral from 1 to tan(5x) of sqrt(1-t) is -2/3 (-tan(5x)+1)^{3/2}

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integral from 1 to tan(5x) of sqrt(1-t)

\int_{1}^{t a n (5 x)} 1 - t d t

- \frac{2}{3} (- tan (5 x) + 1)^{\frac{3}{2}}

Solution steps

\int_{1}^{t a n (5 x)} 1 - t d t

Apply u-substitution

= \int_{0}^{1 - t a n (5 x)} - u d u

Take the constant out:

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

= - \int_{0}^{1 - t a n (5 x)} u d u

Apply the Power Rule

= - [\frac{2}{3} u^{\frac{3}{2}}]_{0}^{1 - t a n (5 x)}

Compute the boundaries:

\frac{2}{3} (- tan (5 x) + 1)^{\frac{3}{2}}

= - \frac{2}{3} (- tan (5 x) + 1)^{\frac{3}{2}}

What is the integral from 1 to tan(5x) of sqrt(1-t) ?
The integral from 1 to tan(5x) of sqrt(1-t) is -2/3 (-tan(5x)+1)^{3/2}