What is the integral from 1 to x of rt^{r-1} ?

The integral from 1 to x of rt^{r-1} is x^r-1

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integral from 1 to x of rt^{r-1}

\int_{1}^{x} r t^{r - 1} d t

x^{r} - 1

Solution steps

\int_{1}^{x} r t^{r - 1} d t

Take the constant out:

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

= r \cdot \int_{1}^{x} t^{- 1 + r} d t

Apply the Power Rule

: [\frac{t ^{r}}{r}]_{1}^{x}

= r [\frac{t ^{r}}{r}]_{1}^{x}

Compute the boundaries:

\frac{x ^{r}}{r} - \frac{1}{r}

= r (\frac{x ^{r}}{r} - \frac{1}{r})

Simplify

= x^{r} - 1

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What is the integral from 1 to x of rt^{r-1} ?
The integral from 1 to x of rt^{r-1} is x^r-1