Resources

Resources

Bytelearn - cat image with glasses

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

int(2x^(3)-4x-8)/((x^(2)-x)(x^(2)+4))dx

$1$ . $\int \frac{2 x^{3}-4 x-8}{\left(x^{2}-x\right)\left(x^{2}+4\right)} d x$

View step-by-step help

View step-by-step help

Find indefinite integrals using the substitution

Full solution

Q.

1

.

\int \frac{2 x^{3}-4 x-8}{\left(x^{2}-x\right)\left(x^{2}+4\right)} d x

Rewrite Integral: Rewrite the integral by factoring the denominator and simplifying the expression. $\newline$ Denominator: $(x^2 - x)(x^2 + 4) = x(x - 1)(x^2 + 4)$ . $\newline$ Attempt partial fraction decomposition: $\newline$ Let $\frac{2x^3 - 4x - 8}{x(x - 1)(x^2 + 4)} = \frac{A}{x} + \frac{B}{x - 1} + \frac{Cx + D}{x^2 + 4}$ . $\newline$ Multiply through by the denominator to clear fractions: $\newline$ $2x^3 - 4x - 8 = A(x - 1)(x^2 + 4) + Bx(x^2 + 4) + (Cx + D)x(x - 1)$ .
Partial Fraction Decomposition: Expand and equate coefficients for $x^3$ , $x^2$ , $x$ , and constant terms. $\newline$ Expanding: $\newline$ $A(x^3 - x^2 + 4x - 4) + B(x^3 + 4x) + Cx^2(x - 1) + Dx(x - 1)$ $\newline$ = $Ax^3 - Ax^2 + 4Ax - 4A + Bx^3 + 4Bx + Cx^3 - Cx^2 + Dx^2 - Dx$ . $\newline$ Combine like terms and equate to the original polynomial: $\newline$ $(A + B + C)x^3 + (-A - C + D)x^2 + (4A + 4B - D)x - 4A = 2x^3 - 4x - 8$ .
Expand and Equate Coefficients: Solve the system of equations for $A$ , $B$ , $C$ , $D$ : $\newline$ $1$ . $A + B + C = 2$ , $\newline$ $2$ . $-A - C + D = 0$ , $\newline$ $3$ . $4A + 4B - D = -4$ , $\newline$ $4$ . $-4A = -8$ . $\newline$ From equation $4$ , $A = 2$ . $\newline$ Substitute $A$ into other equations and solve for $B$ , $C$ , $D$ .
Solve System of Equations: Substituting $A = 2$ into equations: $\newline$ $1$ . $2 + B + C = 2$ , $B + C = 0$ , $\newline$ $2$ . $-2 - C + D = 0$ , $D = 2 + C$ , $\newline$ $3$ . $8 + 4B - D = -4$ . $\newline$ Substitute $D = 2 + C$ into equation $3$ : $\newline$ $8 + 4B - (2 + C) = -4$ , $\newline$ $4B - C = -10$ .
Solve for $B$ and $C$ : Solve $B + C = 0$ and $4B - C = -10$ simultaneously: $\newline$ From $B + C = 0$ , $C = -B$ . $\newline$ Substitute $C = -B$ into $4B - C = -10$ : $\newline$ $4B + B = -10$ , $\newline$ $5B = -10$ , $\newline$ $C$ $0$ . $\newline$ Then $C$ $1$ .
Find D: Find D using $D = 2 + C$ : $D = 2 + 2 = 4$ . Now we have $A = 2$ , $B = -2$ , $C = 2$ , $D = 4$ . The partial fractions are: $\frac{2}{x} - \frac{2}{x - 1} + \frac{2x + 4}{x^2 + 4}$ .
Integrate Each Term: Integrate each term separately: $\newline$ $\int \frac{2}{x} dx = 2 \ln|x| + C_1,$ $\newline$ $\int \frac{-2}{x - 1} dx = -2 \ln|x - 1| + C_2,$ $\newline$ $\int \frac{2x + 4}{x^2 + 4} dx = \ln(x^2 + 4) + 4 \arctan\left(\frac{x}{2}\right) + C_3.$ $\newline$ Combine constants: $C = C_1 + C_2 + C_3.$

More problems from Find indefinite integrals using the substitution

Question

Evaluate the integral.

\newline

\int-x 4^{-3 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int 6 x^{3} e^{-2 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int x 5^{3 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-2 x e^{2 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-2 x \cos (-2 x) d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-x \cos (-3 x) d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int 6 x^{2} 5^{3 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-6 x 4^{-4 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int 6 x^{2} 2^{3 x} d x

\newline

Posted 2 months ago

Question

Evaluate the integral.

\newline

\int-3 x \sin (-2 x) d x

\newline

Posted 2 months ago