Solve ∫ 2/sqrt{(2009)^2+2*2009*213+(213)^2}

$$\int \frac { 2 } { \sqrt { ( 2009 ) ^ { 2 } + 2 \times 2009 \times 213 + ( 213 ) ^ { 2 } } }$$

Evaluate

$$\frac{x}{1111}+С$$

Solution Steps

Calculate 2009 to the power of 2 and get 4036081.

$$\int \frac{2}{\sqrt{4036081+2\times 2009\times 213+213^{2}}}\mathrm{d}x$$

Multiply 2 and 2009 to get 4018.

$$\int \frac{2}{\sqrt{4036081+4018\times 213+213^{2}}}\mathrm{d}x$$

Multiply 4018 and 213 to get 855834.

$$\int \frac{2}{\sqrt{4036081+855834+213^{2}}}\mathrm{d}x$$

Add 4036081 and 855834 to get 4891915.

$$\int \frac{2}{\sqrt{4891915+213^{2}}}\mathrm{d}x$$

Calculate 213 to the power of 2 and get 45369.

$$\int \frac{2}{\sqrt{4891915+45369}}\mathrm{d}x$$

Add 4891915 and 45369 to get 4937284.

$$\int \frac{2}{\sqrt{4937284}}\mathrm{d}x$$

Calculate the square root of 4937284 and get 2222.

$$\int \frac{2}{2222}\mathrm{d}x$$

Reduce the fraction \frac{2}{2222} to lowest terms by extracting and canceling out 2.

$$\int \frac{1}{1111}\mathrm{d}x$$

Find the integral of \frac{1}{1111} using the table of common integrals rule \int a\mathrm{d}x=ax.

$$\frac{x}{1111}$$

If F\left(x\right) is an antiderivative of f\left(x\right), then the set of all antiderivatives of f\left(x\right) is given by F\left(x\right)+C. Therefore, add the constant of integration C\in \mathrm{R} to the result.

$$\frac{x}{1111}+С$$

Differentiate w.r.t. x

$$\frac{1}{1111} = 0.0009000900090009$$