What is the integral from 1 to 7 of (1-x)(x-7) ?

The integral from 1 to 7 of (1-x)(x-7) is 36

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integral from 1 to 7 of (1-x)(x-7)

\int_{1}^{7} (1 - x) (x - 7) d x

36

Solution steps

\int_{1}^{7} (1 - x) (x - 7) d x

Expand

(1 - x) (x - 7) : 8 x - x^{2} - 7

= \int_{1}^{7} 8 x - x^{2} - 7 d x

Apply the Sum Rule:

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

= \int_{1}^{7} 8 x d x - \int_{1}^{7} x^{2} d x - \int_{1}^{7} 7 d x

\int_{1}^{7} 8 x d x = 192

\int_{1}^{7} x^{2} d x = 114

\int_{1}^{7} 7 d x = 42

= 192 - 114 - 42

Simplify

= 36

\int_{0}^{2} \frac{1}{4 x + 1} d x

\int_{2}^{3} \frac{1}{( x - 1 ) ^{2}} d x

\int_{3}^{7} \frac{3}{7 - x} d x

\int_{0}^{1} cos (t) d t

\int_{0}^{3} ∣ 5 x - 9 ∣ d x

What is the integral from 1 to 7 of (1-x)(x-7) ?
The integral from 1 to 7 of (1-x)(x-7) is 36