Evaluate int_(3)^(4)(2x^(2)-15 x+22)/(x-5)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Question

Evaluate  int_(3)^(4)(2x^(2)-15 x+22)/(x-5)dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Bytelearn · Accepted Answer

Perform Polynomial Long Division: We need to evaluate the definite integral of the rational function (2x^2 - 15x + 22) / (x - 5) from x = 3 to x = 4. To do this, we will first perform polynomial long division to simplify the integrand.Split Integral into Two Parts: Performing the polynomial long division of (2x^2 - 15x + 22) by (x - 5), we get:  2x^2 - 15x + 22 = (x - 5)(2x + 5) + (-3x + 47)  So, the integral becomes:  int_(3)^(4)(2x + 5 - (3x - 47)/(x - 5))dxIntegrate Polynomial Part: Now we can split the integral into two parts:  int_(3)^(4)(2x + 5)dx - int_(3)^(4)((3x - 47)/(x - 5))dx  We will integrate each part separately.Integrate Rational Part: First, we integrate the polynomial part:  int_(3)^(4)(2x + 5)dx = [x^2 + 5x]_(3)^(4)  Evaluating from x = 3 to x = 4, we get:  (4^2 + 5*4) - (3^2 + 5*3) = (16 + 20) - (9 + 15) = 36 - 24 = 12Substitution Method: Next, we integrate the rational part:  int_(3)^(4)((3x - 47)/(x - 5))dx  To integrate this, we will use the substitution method. Let u = x - 5, then du = dx and x = u + 5.  The limits of integration also change: when x = 3, u = -2; when x = 4, u = -1.Simplify the Integrand: Substituting x with u + 5, the integral becomes:  int_(-2)^(-1)((3(u + 5) - 47)/u)du  Simplify the integrand:  int_(-2)^(-1)((3u + 15 - 47)/u)du = int_(-2)^(-1)((3u - 32)/u)du  Split the integral:  int_(-2)^(-1)(3 - 32/u)duIntegrate Each Term: Now we integrate each term separately:  int_(-2)^(-1)3du - int_(-2)^(-1)(32/u)du  The first integral is straightforward:  int_(-2)^(-1)3du = [3u]_(-2)^(-1) = 3(-1) - 3(-2) = -3 + 6 = 3  The second integral involves a natural logarithm:  int_(-2)^(-1)(32/u)du = 32 * ln|u| evaluated from u = -2 to u = -1.Evaluate Logarithmic Part: Evaluating the logarithmic part, we get:  32 * ln|u|_(-2)^(-1) = 32(ln|-1| - ln|-2|) = 32(ln(1) - ln(2))  Since ln(1) = 0, this simplifies to:  -32 * ln(2)Combine Results: Combining the results from the polynomial and rational parts, we get the final value of the definite integral:  12 (from the polynomial part) - 32 * ln(2) (from the rational part)  This is the final answer in its simplest form with the logarithm condensed.

Evaluate $\int_{3}^{4} \frac{2 x^{2}-15 x+22}{x-5} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Full solution

More problems from Find indefinite integrals using the substitution

Related topics

Evaluate ∫342x2−15x+22x−5dx \int_{3}^{4} \frac{2 x^{2}-15 x+22}{x-5} d x ∫34​x−52x2−15x+22​dx. Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).

Evaluate $\int_{3}^{4} \frac{2 x^{2}-15 x+22}{x-5} d x$ . Write your answer in simplest form with all logs condensed into a single logarithm (if necessary).