What is the integral of 25-(t-5)^2 ?

The integral of 25-(t-5)^2 is -(t^3)/3+5t^2+C

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integral of 25-(t-5)^2

\int 25 - (t - 5)^{2} d t

- \frac{t ^{3}}{3} + 5 t^{2} + C

Solution steps

\int 25 - (t - 5)^{2} d t

Apply the Sum Rule:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int 25 d t - \int (t - 5)^{2} d t

\int 25 d t = 25 t

\int (t - 5)^{2} d t = \frac{t ^{3}}{3} - 5 t^{2} + 25 t

= 25 t - (\frac{t ^{3}}{3} - 5 t^{2} + 25 t)

Simplify

= - \frac{t ^{3}}{3} + 5 t^{2}

Add a constant to the solution

= - \frac{t ^{3}}{3} + 5 t^{2} + C

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What is the integral of 25-(t-5)^2 ?
The integral of 25-(t-5)^2 is -(t^3)/3+5t^2+C