What is the integral from-1 to 3 of 5xsqrt(x+1) ?

The integral from-1 to 3 of 5xsqrt(x+1) is 112/3

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integral from-1 to 3 of 5xsqrt(x+1)

\int_{- 1}^{3} 5 x x + 1 d x

\frac{112}{3}

Solution steps

\int_{- 1}^{3} 5 x x + 1 d x

Take the constant out:

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

= 5 \cdot \int_{- 1}^{3} x x + 1 d x

Apply u-substitution

: \int_{0}^{4} (u - 1) u d u

= 5 \cdot \int_{0}^{4} (u - 1) u d u

Expand

(u - 1) u : u^{\frac{3}{2}} - u

= 5 \cdot \int_{0}^{4} u^{\frac{3}{2}} - u d u

Apply the Sum Rule:

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

= 5 (\int_{0}^{4} u^{\frac{3}{2}} d u - \int_{0}^{4} u d u)

\int_{0}^{4} u^{\frac{3}{2}} d u = \frac{64}{5}

\int_{0}^{4} u d u = \frac{16}{3}

= 5 (\frac{64}{5} - \frac{16}{3})

Simplify

5 (\frac{64}{5} - \frac{16}{3}) : \frac{112}{3}

= \frac{112}{3}

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What is the integral from-1 to 3 of 5xsqrt(x+1) ?
The integral from-1 to 3 of 5xsqrt(x+1) is 112/3