What is the integral of (2t)/(sqrt(25-t^2)) ?

The integral of (2t)/(sqrt(25-t^2)) is -2sqrt(25-t^2)+C

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integral of (2t)/(sqrt(25-t^2))

\int \frac{2 t}{25 - t ^{2}} d t

- 2 25 - t^{2} + C

Solution steps

\int \frac{2 t}{25 - t ^{2}} d t

Take the constant out:

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

= 2 \cdot \int \frac{t}{25 - t ^{2}} d t

Apply u-substitution

= 2 \cdot \int - \frac{1}{2 u} d u

Take the constant out:

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

= 2 (- \frac{1}{2} \cdot \int \frac{1}{u} d u)

Apply the Power Rule

= 2 (- \frac{1}{2} \cdot 2 u^{\frac{1}{2}})

Substitute back

u = 25 - t^{2}

= 2 (- \frac{1}{2} \cdot 2 (25 - t^{2})^{\frac{1}{2}})

Simplify

2 (- \frac{1}{2} \cdot 2 (25 - t^{2})^{\frac{1}{2}}) : - 2 25 - t^{2}

= - 2 25 - t^{2}

Add a constant to the solution

= - 2 25 - t^{2} + C

What is the integral of (2t)/(sqrt(25-t^2)) ?
The integral of (2t)/(sqrt(25-t^2)) is -2sqrt(25-t^2)+C