Algebra
Calculus
Trigonometry
Matrix
Question
Evaluate the integral
−51ln(∣x∣)−59ln(∣x−5∣)+C,C∈R
Evaluate
∫(x×1)(x−5)1−2xdx
Remove the parentheses
∫x×1×(x−5)1−2xdx
Multiply the terms
More Steps
Multiply the terms
x×1×(x−5)
Rewrite the expression
x(x−5)
Apply the distributive property
x×x−x×5
Multiply the terms
x2−x×5
Use the commutative property to reorder the terms
x2−5x
∫x2−5x1−2xdx
Rewrite the fraction
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Evaluate
x2−5x1−2x
Factor the expression
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Evaluate
x2−5x
Rewrite the expression
x×x−x×5
Factor out x from the expression
x(x−5)
x(x−5)1−2x
For each factor in the denominator,write a new fraction
x?+x−5?
Write the terms in the numerator
xA+x−5B
Set the sum of fractions equal to the original fraction
x(x−5)1−2x=xA+x−5B
Multiply both sides
x(x−5)1−2x×x(x−5)=xA×x(x−5)+x−5B×x(x−5)
Simplify the expression
1−2x=(x−5)A+xB
Simplify the expression
1−2x=xA−5A+xB
Group the terms
1−2x=(A+B)x−5A
Equate the coefficients
{−2=A+B1=−5A
Swap the sides
{A+B=−2−5A=1
Solve the equation for A
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Evaluate
−5A=1
Change the signs on both sides of the equation
5A=−1
Divide both sides
55A=5−1
Divide the numbers
A=5−1
Use b−a=−ba=−ba to rewrite the fraction
A=−51
{A+B=−2A=−51
Substitute the given value of A into the equation A+B=−2
−51+B=−2
Move the constant to the right-hand side and change its sign
B=−2+51
Add the numbers
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Evaluate
−2+51
Write all numerators above the least common denominator 5
−1×52×5+51
Calculate
−510+51
Add the terms
5−10+1
Add the terms
5−9
Rewrite the fraction
−59
B=−59
Calculate
{A=−51B=−59
Substitute back
−5x1−5x−259
∫(−5x1−5x−259)dx
Use the property of integral ∫f(x)±g(x)dx=∫f(x)dx±∫g(x)dx
∫−5x1dx+∫−5x−259dx
Evaluate the integral
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Evaluate
∫−5x1dx
Rewrite the expression
∫−51×x1dx
Use the property of integral ∫kf(x)dx=k∫f(x)dx
−51×∫x1dx
Use the property of integral ∫x1dx=ln∣x∣
−51ln(∣x∣)
−51ln(∣x∣)+∫−5x−259dx
Evaluate the integral
More Steps
Evaluate
∫−5x−259dx
Rewrite the expression
∫−59×x−51dx
Use the property of integral ∫kf(x)dx=k∫f(x)dx
−59×∫x−51dx
Use the property of integral ∫ax+b1dx=a1ln∣ax+b∣
−59ln(∣x−5∣)
−51ln(∣x∣)−59ln(∣x−5∣)
Solution
−51ln(∣x∣)−59ln(∣x−5∣)+C,C∈R
Show Solution
