What is the integral of 7/(26+x^2-10x) ?

The integral of 7/(26+x^2-10x) is 7arctan(x-5)+C

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integral of 7/(26+x^2-10x)

\int \frac{7}{26 + x ^{2} - 10 x} d x

7 arctan (x - 5) + C

Solution steps

\int \frac{7}{26 + x ^{2} - 10 x} d x

Take the constant out:

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

= 7 \cdot \int \frac{1}{26 + x ^{2} - 10 x} d x

Complete the square

26 + x^{2} - 10 x : (x - 5)^{2} + 1

= 7 \cdot \int \frac{1}{( x - 5 ) ^{2} + 1} d x

Apply u-substitution

: \int \frac{1}{u ^{2} + 1} d u

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

Use the common integral:

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

= 7 arctan (u)

Substitute back

u = x - 5

= 7 arctan (x - 5)

Add a constant to the solution

= 7 arctan (x - 5) + C

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What is the integral of 7/(26+x^2-10x) ?
The integral of 7/(26+x^2-10x) is 7arctan(x-5)+C