What is the integral from 7 to 8 of 7/(50+x^2-14x) ?

The integral from 7 to 8 of 7/(50+x^2-14x) is (7pi)/4

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integral from 7 to 8 of 7/(50+x^2-14x)

\int_{7}^{8} \frac{7}{50 + x ^{2} - 14 x} d x

\frac{7 π}{4}

Decimal Notation

5.49778 \dots

Solution steps

\int_{7}^{8} \frac{7}{50 + x ^{2} - 14 x} d x

Take the constant out:

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

= 7 \cdot \int_{7}^{8} \frac{1}{50 + x ^{2} - 14 x} d x

Complete the square

50 + x^{2} - 14 x : (x - 7)^{2} + 1

= 7 \cdot \int_{7}^{8} \frac{1}{( x - 7 ) ^{2} + 1} d x

Apply u-substitution

: \int_{0}^{1} \frac{1}{u ^{2} + 1} d u

= 7 \cdot \int_{0}^{1} \frac{1}{u ^{2} + 1} d u

Use the common integral:

\int \frac{1}{u ^{2} + 1} d u = arctan (u)

= 7 [arctan (u)]_{0}^{1}

Compute the boundaries:

\frac{π}{4}

= 7 \cdot \frac{π}{4}

Simplify

= \frac{7 π}{4}

\int 2 x^{\frac{5}{2}} d x

x \to 5 lim (f (x))

y = x^{7 x}

e^{- 2 x}, 2

\frac{d y}{d x} = \frac{x}{y}, y (0) = - 3

What is the integral from 7 to 8 of 7/(50+x^2-14x) ?
The integral from 7 to 8 of 7/(50+x^2-14x) is (7pi)/4