What is the integral of 7/(x^2+25) ?

The integral of 7/(x^2+25) is 7/5 arctan(x/5)+C

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integral of 7/(x^2+25)

\int \frac{7}{x ^{2} + 25} d x

\frac{7}{5} arctan (\frac{x}{5}) + C

Solution steps

\int \frac{7}{x ^{2} + 25} 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}{x ^{2} + 25} d x

Apply Integral Substitution

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

Take the constant out:

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

= 7 \cdot \frac{1}{5} \cdot \int \frac{1}{u ^{2} + 1} d u

Use the common integral:

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

= 7 \cdot \frac{1}{5} arctan (u)

Substitute back

u = \frac{x}{5}

= 7 \cdot \frac{1}{5} arctan (\frac{x}{5})

Simplify

7 \cdot \frac{1}{5} arctan (\frac{x}{5}) : \frac{7}{5} arctan (\frac{x}{5})

= \frac{7}{5} arctan (\frac{x}{5})

Add a constant to the solution

= \frac{7}{5} arctan (\frac{x}{5}) + C

\frac{d}{d x} (4 x^{\frac{2}{3}} + x^{\frac{5}{3}})

f (x) = x^{3} - 2 x, at x = 2

\frac{d y}{d x} + 5 (tan (5 x)) y = 3 cos (5 x)

\int 60 x^{2} (1 - 4 x^{3})^{4} d x

What is the integral of 7/(x^2+25) ?
The integral of 7/(x^2+25) is 7/5 arctan(x/5)+C