What is the integral from 1 to sin(x) of 3t^2 ?

The integral from 1 to sin(x) of 3t^2 is sin^3(x)-1

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integral from 1 to sin(x) of 3t^2

\int_{1}^{s i n (x)} 3 t^{2} d t

sin^{3} (x) - 1

Solution steps

\int_{1}^{s i n (x)} 3 t^{2} d t

Take the constant out:

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

= 3 \cdot \int_{1}^{s i n (x)} t^{2} d t

Apply the Power Rule

: [\frac{t ^{3}}{3}]_{1}^{s i n (x)}

= 3 [\frac{t ^{3}}{3}]_{1}^{s i n (x)}

Compute the boundaries:

\frac{sin ^{3} ( x )}{3} - \frac{1}{3}

= 3 (\frac{sin ^{3} ( x )}{3} - \frac{1}{3})

Simplify

= sin^{3} (x) - 1

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What is the integral from 1 to sin(x) of 3t^2 ?
The integral from 1 to sin(x) of 3t^2 is sin^3(x)-1