What is the integral of (2+5x)/(sqrt(1+4x+5x^2)) ?

The integral of (2+5x)/(sqrt(1+4x+5x^2)) is sqrt(5x^2+4x+1)+C

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integral of (2+5x)/(sqrt(1+4x+5x^2))

\int \frac{2 + 5 x}{1 + 4 x + 5 x ^{2}} d x

5 x^{2} + 4 x + 1 + C

Solution steps

\int \frac{2 + 5 x}{1 + 4 x + 5 x ^{2}} d x

Apply u-substitution

: \int \frac{u}{5 u ^{2} + 1} d u

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

Take the constant out:

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

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

Apply u-substitution

: \int \frac{1}{2 v} d v

= \frac{1}{5} \cdot \int \frac{1}{2 v} d v

Take the constant out:

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

= \frac{1}{5} \cdot \frac{1}{2} \cdot \int \frac{1}{v} d v

Apply the Power Rule

: 2 v^{\frac{1}{2}}

= \frac{1}{5} \cdot \frac{1}{2} \cdot 2 v^{\frac{1}{2}}

Substitute back

= \frac{1}{5} \cdot \frac{1}{2} \cdot 2 ((2 + 5 x)^{2} + 1)^{\frac{1}{2}}

Simplify

\frac{1}{5} \cdot \frac{1}{2} \cdot 2 ((2 + 5 x)^{2} + 1)^{\frac{1}{2}} : 5 x^{2} + 4 x + 1

= 5 x^{2} + 4 x + 1

Add a constant to the solution

= 5 x^{2} + 4 x + 1 + C

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What is the integral of (2+5x)/(sqrt(1+4x+5x^2)) ?
The integral of (2+5x)/(sqrt(1+4x+5x^2)) is sqrt(5x^2+4x+1)+C