What is the integral from 3 to 9 of 7arctan(x) ?

The integral from 3 to 9 of 7arctan(x) is 7(9arctan(9)-3arctan(3)+(ln(5/41))/2)

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integral from 3 to 9 of 7arctan(x)

\int_{3}^{9} 7 arctan (x) d x

7 (9 arctan (9) - 3 arctan (3) + \frac{ln ( \frac{5}{41} )}{2})

Decimal Notation

58.39433 \dots

Solution steps

\int_{3}^{9} 7 arctan (x) d x

Take the constant out:

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

= 7 \cdot \int_{3}^{9} arctan (x) d x

Apply Integration By Parts

: [x arctan (x) - \int \frac{x}{x ^{2} + 1} d x]_{3}^{9}

= 7 [x arctan (x) - \int \frac{x}{x ^{2} + 1} d x]_{3}^{9}

\int \frac{x}{x ^{2} + 1} d x = \frac{1}{2} ln x^{2} + 1

= 7 [x arctan (x) - \frac{1}{2} ln x^{2} + 1]_{3}^{9}

Compute the boundaries:

9 arctan (9) - 3 arctan (3) + \frac{- ln ( 82 ) + ln ( 10 )}{2}

= 7 (9 arctan (9) - 3 arctan (3) + \frac{- ln ( 82 ) + ln ( 10 )}{2})

Simplify

= 7 (9 arctan (9) - 3 arctan (3) + \frac{ln ( \frac{5}{41} )}{2})

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What is the integral from 3 to 9 of 7arctan(x) ?
The integral from 3 to 9 of 7arctan(x) is 7(9arctan(9)-3arctan(3)+(ln(5/41))/2)