Question

( 2ax+x^2)dy/dx = a²+2ax
​

Answer 1

Answer:

the numerator were the derivative of the denominator, we

would be able to integrate using int%28du%2Fu%29=ln%28abs%28u%29%29%2BC

The derivative of the denominator is 2a+2x.

The numerator a+2x only needs to have "a" added to it to become that

So in the numerator we will add "a" and subtract "a"

y%22%22=%22%22a%2Aint%28expr%28%28red%28a%29%2Ba%2B2x-red%28a%29%29%2F%282ax%2Bx%5E2%29%29dx%29

y%22%22=%22%22a%2Aint%28expr%28%282a%2B2x-red%28a%29%29%2F%282ax%2Bx%5E2%29%29dx%29

Write as the difference of two fractions and integrate each:

y%22%22=%22%22

y%22%22=%22%22

y%22%22=%22%22a%2Aint%28expr%28%282a%2B2x%29%2F%282ax%2Bx%5E2%29%29dx%29%22%22-%22%22a%2Aint%28expr%28a%2F%282ax%2Bx%5E2%29%29dx%29

Use the natural log formula on the first integral, and take the

constant "a" outside the second integral:

y%22%22=%22%22a%2Aln%28abs%282ax%2Bx%5E2%29%29%22%22-%22%22a%5E2%2Aint%28expr%281%2F%282ax%2Bx%5E2%29%29dx%29

To the side, let's complete the square in the denominator by

adding and subtrracting a2:

2ax%2Bx%5E2%22%22=%22%22x%5E2%2B2ax%2Ba%5E2-a%5E2%22%22=%22%22%28x%2Ba%29%5E2-a%5E2

y%22%22=%22%22a%2Aln%28abs%282ax%2Bx%5E2%29%29%22%22-%22%22a%5E2%2Aint%28expr%281%2F%28%28x%2Ba%29%5E2-a%5E2%29%29dx%29

To do the remaining integral we use the formula int%28+du%2F%28u%5E2-a%5E2%29+%29%22%22=%22%22expr%28+1%2F%282a%29+%29ln%28abs%28+%28u-a%29%2F%28u%2Ba%29%29+%29%2BC

y%22%22=%22%22a%2Aln%28abs%282ax%2Bx%5E2%29%29%22%22-%22%22

y%22%22=%22%22a%2Aln%28abs%282ax%2Bx%5E2%29%29%22%22-%22%22expr%28+a%2F2+%29ln%28abs%28+x%2F%28x%2B2a%29%29+%29%2BC

Edwin