What is the integral of (16x^2)/(16x^2+25) ?

The integral of (16x^2)/(16x^2+25) is 1/4 (4x-5arctan(4/5 x))+C

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integral of (16x^2)/(16x^2+25)

\int \frac{16 x ^{2}}{16 x ^{2} + 25} d x

\frac{1}{4} (4 x - 5 arctan (\frac{4}{5} x)) + C

Solution steps

\int \frac{16 x ^{2}}{16 x ^{2} + 25} d x

Take the constant out:

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

= 16 \cdot \int \frac{x ^{2}}{16 x ^{2} + 25} d x

16 x^{2} + 25 = 25 (\frac{16 x ^{2}}{25} + 1)

= 16 \cdot \int \frac{x ^{2}}{25 ( \frac{16 x ^{2}}{25} + 1 )} d x

Take the constant out:

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

= 16 \cdot \frac{1}{25} \cdot \int \frac{x ^{2}}{\frac{16 x ^{2}}{25} + 1} d x

Apply Integral Substitution

= 16 \cdot \frac{1}{25} \cdot \int \frac{125 u ^{2}}{64 ( u ^{2} + 1 )} d u

Take the constant out:

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

= 16 \cdot \frac{1}{25} \cdot \frac{125}{64} \cdot \int \frac{u ^{2}}{u ^{2} + 1} d u

\frac{u ^{2}}{u ^{2} + 1} = - \frac{1}{u ^{2} + 1} + 1

= 16 \cdot \frac{1}{25} \cdot \frac{125}{64} \cdot \int - \frac{1}{u ^{2} + 1} + 1 d u

Apply the Sum Rule:

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

= 16 \cdot \frac{1}{25} \cdot \frac{125}{64} (- \int \frac{1}{u ^{2} + 1} d u + \int 1 d u)

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

\int 1 d u = u

= 16 \cdot \frac{1}{25} \cdot \frac{125}{64} (- arctan (u) + u)

Substitute back

u = \frac{4}{5} x

= 16 \cdot \frac{1}{25} \cdot \frac{125}{64} (- arctan (\frac{4}{5} x) + \frac{4}{5} x)

Simplify

16 \cdot \frac{1}{25} \cdot \frac{125}{64} (- arctan (\frac{4}{5} x) + \frac{4}{5} x) : \frac{1}{4} (4 x - 5 arctan (\frac{4}{5} x))

= \frac{1}{4} (4 x - 5 arctan (\frac{4}{5} x))

Add a constant to the solution

= \frac{1}{4} (4 x - 5 arctan (\frac{4}{5} x)) + C

\int \frac{x ^{2} + 4 x - 1}{x ^{3} - x} d x

f (x) = arctan (\frac{1 - x}{1 + x})

\frac{x}{7 + x}

\frac{d}{d x} (e^{(x + l n (x))})

y = sin (x), y = sin (2 x), x = 0, x = \frac{π}{3}

What is the integral of (16x^2)/(16x^2+25) ?
The integral of (16x^2)/(16x^2+25) is 1/4 (4x-5arctan(4/5 x))+C