int(6x-11)/((x-1)^(2))dx

Identify Integral: Identify the integral to be solved. We need to find the integral of the function (6x-11)/((x-1)^(2)) with respect to x.Simplify Integrand: Simplify the integrand if possible. The integrand is already in a simplified form, so we can proceed to the next step.Partial Fraction Decomposition: Use partial fraction decomposition if applicable. Since the denominator is a perfect square, we can try to decompose the numerator in a way that will allow us to integrate easily. We can write the integrand as A/(x-1) + B/((x-1)^2) and solve for A and B.Find A and B: Find the values of A and B. Multiplying both sides by (x-1)^2, we get: 6x - 11 = A(x-1) + B Now, let's find A and B by plugging in convenient values for x.Plug in Values: Plug in x = 1 to find B. If we plug in x = 1, we get: 6(1) - 11 = A(1-1) + B -5 = B So, B = -5.Differentiate for A: Differentiate both sides with respect to x to find A. Differentiating both sides of the equation 6x - 11 = A(x-1) + B with respect to x gives us: 6 = A So, A = 6.Rewrite Integral: Rewrite the integral using the values of A and B. Now that we have A = 6 and B = -5, we can rewrite the integral as: ∫(6x-11)/((x-1)^(2))dx = ∫6/(x-1)dx - ∫5/((x-1)^2)dxIntegrate Terms: Integrate each term separately. The integral of 6/(x-1)dx is 6ln|x-1|, and the integral of 5/((x-1)^2)dx is -5/(x-1).Combine Results: Combine the results and add the constant of integration. The final answer is the sum of the two integrals plus the constant of integration C: ∫(6x-11)/((x-1)^(2))dx = 6ln|x-1| - 5/(x-1) + C

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$\int\frac{6x-11}{(x-1)^{2}}dx$

View step-by-step help

Home

Math Problems

Calculus

Evaluate definite integrals using the chain rule

Full solution

\int\frac{6x-11}{(x-1)^{2}}dx

Identify Integral: Identify the integral to be solved. $\newline$ We need to find the integral of the function $(6x-11)/((x-1)^{2})$ with respect to $x$ .
Simplify Integrand: Simplify the integrand if possible. $\newline$ The integrand is already in a simplified form, so we can proceed to the next step.
Partial Fraction Decomposition: Use partial fraction decomposition if applicable. $\newline$ Since the denominator is a perfect square, we can try to decompose the numerator in a way that will allow us to integrate easily. We can write the integrand as $\frac{A}{x-1}$ + $\frac{B}{(x-1)^2}$ and solve for $A$ and $B$ .
Find $A$ and $B$ : Find the values of $A$ and $B$ . $\newline$ Multiplying both sides by $(x-1)^2$ , we get: $\newline$ $6x - 11 = A(x-1) + B$ $\newline$ Now, let's find $A$ and $B$ by plugging in convenient values for $x$ .
Plug in Values: Plug in $x = 1$ to find $B$ . If we plug in $x = 1$ , we get: $6(1) - 11 = A(1-1) + B$ $-5 = B$ So, $B = -5$ .
Differentiate for $A$ : Differentiate both sides with respect to $x$ to find $A$ . Differentiating both sides of the equation $6x - 11 = A(x-1) + B$ with respect to $x$ gives us: $6 = A$ So, $A = 6$ .
Rewrite Integral: Rewrite the integral using the values of $A$ and $B$ . Now that we have $A = 6$ and $B = -5$ , we can rewrite the integral as: $\int(6x-11)/((x-1)^{2})dx = \int 6/(x-1)dx - \int 5/((x-1)^2)dx$
Integrate Terms: Integrate each term separately. $\newline$ The integral of $\frac{6}{x-1}\,dx$ is $6\ln|x-1|$ , and the integral of $\frac{5}{(x-1)^2}\,dx$ is $-\frac{5}{x-1}$ .
Combine Results: Combine the results and add the constant of integration. The final answer is the sum of the two integrals plus the constant of integration $C$ : $\int(6x-11)/((x-1)^{2})dx = 6\ln|x-1| - \frac{5}{x-1} + C$