What is the integral from 3 to 9 of 20e^{-0.1x^2} ?

The integral from 3 to 9 of 20e^{-0.1x^2} is 10.06967…

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integral from 3 to 9 of 20e^{-0.1x^2}

\int_{3}^{9} 20 e^{- 0.1 x^{2}} d x

10.06967 \dots

Solution steps

\int_{3}^{9} 20 e^{- 0.1 x^{2}} d x

Take the constant out:

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

= 20 \cdot \int_{3}^{9} e^{- 0.1 x^{2}} d x

= 20 \cdot \int_{3}^{9} e^{- (0.1 x)^{2}} d x

Apply u-substitution

: \int_{0.94868 \dots}^{2.84604 \dots} e^{- u^{2}} \frac{1}{0.31622 \dots} d u

= 20 \cdot \int_{0.94868 \dots}^{2.84604 \dots} e^{- u^{2}} \frac{1}{0.31622 \dots} d u

Take the constant out:

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

= 20 \cdot \frac{1}{0.31622 \dots} \cdot \int_{0.94868 \dots}^{2.84604 \dots} e^{- u^{2}} d u

Simplify

= 20 \cdot 3.16227 \dots \cdot \int_{0.94868 \dots}^{2.84604 \dots} e^{- u^{2}} d u

Use the nonelementary integral:

\int e^{- u^{2}} d u = \frac{π}{2} erf (u)

= 20 \cdot 3.16227 \dots [\frac{π}{2} erf (u)]_{0.94868 \dots}^{2.84604 \dots}

Simplify

= 63.24555 \dots [0.88622 \dots erf (u)]_{0.94868 \dots}^{2.84604 \dots}

Compute the boundaries:

0.15921 \dots

= 63.24555 \dots \cdot 0.15921 \dots

Simplify

= 10.06967 \dots

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What is the integral from 3 to 9 of 20e^{-0.1x^2} ?
The integral from 3 to 9 of 20e^{-0.1x^2} is 10.06967…