Evaluate the following
(x+7)dx/x^2+14x+55
makes no sense as written
better look again and copy it accurately
If (x+7)dx/x^2+14x+55
mean
( x + 7 ) dx / ( x ^ 2 + 14 x + 55 )
then substitute :
u = x ^ 2 + 14 x + 5
d u = ( 2 x + 14 ) dx = 2 ( x + 7 ) d x Divide both sides by 2
du / 2 = ( x + 7 ) d x
( x + 7 ) d x = d u / 2
integral of [ ( x + 7 ) dx / ( x ^ 2 + 14 x + 55 ) ] =
integral of [ ( d u / 2 ) / u ) ] =
( 1 / 2 ) integral of ( du / u ) =
( 1 / 2 ) log ( u ) + C =
( 1 / 2 ) log ( x ^ 2 + 14 x + 5 ) + C =
log [ sqrt ( x ^ 2 + 14 x + 5 ) ] + C
Remark :
log is the natural logarithm
( 1 / n ) log ( a ) = log ] ( n - th root ( a ) ]
( 1 / 2 ) log ( a ) = log [ sqrt ( a ) ]
To evaluate the integral ∫ (x+7)dx / (x^2 + 14x + 55), you can use a method known as partial fraction decomposition. This method involves expressing the rational function as a sum of simpler fractions.
1. Start by factoring the denominator: x^2 + 14x + 55 = (x + 5)(x + 11).
2. Since the denominator is a quadratic expression, it can be factored into linear terms. We can express the rational function as follows:
(x + 7) / (x^2 + 14x + 55) = A / (x + 5) + B / (x + 11)
3. Find the values of A and B by equating the numerators:
(x + 7) = A(x + 11) + B(x + 5)
4. Expand the equation:
x + 7 = Ax + 11A + Bx + 5B
5. Group the terms with the same powers of x:
x + 7 = (A + B)x + (11A + 5B)
6. Equate the coefficients of corresponding powers of x:
Coefficient of x: 1 = A + B
Constant term: 7 = 11A + 5B
7. Solve the system of equations to find the values of A and B.
From the first equation, A = 1 - B.
Substituting this into the second equation:
7 = 11(1 - B) + 5B
7 = 11 - 11B + 5B
7 = 11 - 6B
6B = 11 - 7
6B = 4
B = 4/6
B = 2/3
Substituting B = 2/3 into the first equation:
A = 1 - B
A = 1 - 2/3
A = 3/3 - 2/3
A = 1/3
Therefore, A = 1/3 and B = 2/3.
8. Rewrite the rational function in terms of the partial fractions:
(x + 7) / (x^2 + 14x + 55) = 1/3 * (1 / (x + 5)) + 2/3 * (1 / (x + 11))
Now, the integral becomes:
∫ (x + 7)dx / (x^2 + 14x + 55) = ∫ (1/3 * (1 / (x + 5)) + 2/3 * (1 / (x + 11))) dx
9. Integrate each term separately:
∫ (1/3 * (1 / (x + 5)) + 2/3 * (1 / (x + 11))) dx = (1/3) * ln| x + 5 | + (2/3) * ln| x + 11 | + C
where ln denotes the natural logarithm and C is the constant of integration.
Therefore, the evaluated integral is:
∫ (x + 7)dx / (x^2 + 14x + 55) = (1/3) * ln| x + 5 | + (2/3) * ln| x + 11 | + C