What is the integral from 1 to a of 1/(3t^3) ?

The integral from 1 to a of 1/(3t^3) is 1/3 (-1/(2a^2)+1/2)

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integral from 1 to a of 1/(3t^3)

\int_{1}^{a} \frac{1}{3 t ^{3}} d t

\frac{1}{3} (- \frac{1}{2 a ^{2}} + \frac{1}{2})

Solution steps

\int_{1}^{a} \frac{1}{3 t ^{3}} d t

Take the constant out:

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

= \frac{1}{3} \cdot \int_{1}^{a} \frac{1}{t ^{3}} d t

Apply the Power Rule

: [- \frac{1}{2 t ^{2}}]_{1}^{a}

= \frac{1}{3} [- \frac{1}{2 t ^{2}}]_{1}^{a}

Compute the boundaries:

- \frac{1}{2 a ^{2}} + \frac{1}{2}

= \frac{1}{3} (- \frac{1}{2 a ^{2}} + \frac{1}{2})

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What is the integral from 1 to a of 1/(3t^3) ?
The integral from 1 to a of 1/(3t^3) is 1/3 (-1/(2a^2)+1/2)