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

The integral of (4x^2+1)/(sqrt(2x+5)) is 4(x^2sqrt(2x+5)-2/15 (3x-5)(2x+5)^{3/2})+sqrt(2x+5)+C

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

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

4 (x^{2} 2 x + 5 - \frac{2}{15} (3 x - 5) (2 x + 5)^{\frac{3}{2}}) + 2 x + 5 + C

Solution steps

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

Expand

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

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

\int \frac{4 x ^{2}}{2 x + 5} d x = 4 (x^{2} 2 x + 5 - \frac{2}{15} (3 x - 5) (2 x + 5)^{\frac{3}{2}})

\int \frac{1}{2 x + 5} d x = 2 x + 5

= 4 (x^{2} 2 x + 5 - \frac{2}{15} (3 x - 5) (2 x + 5)^{\frac{3}{2}}) + 2 x + 5

Add a constant to the solution

= 4 (x^{2} 2 x + 5 - \frac{2}{15} (3 x - 5) (2 x + 5)^{\frac{3}{2}}) + 2 x + 5 + C

\int e^{- \frac{3}{2} t} d t

3 x^{2} + 4 x + 8

(x^{2} + 7)^{5}

x, x^{3} - x^{2} - x

2 sin (x), 4 cos (x), [0, 0.6 π]

What is the integral of (4x^2+1)/(sqrt(2x+5)) ?
The integral of (4x^2+1)/(sqrt(2x+5)) is 4(x^2sqrt(2x+5)-2/15 (3x-5)(2x+5)^{3/2})+sqrt(2x+5)+C