Evaluate the integral. int(x-1)(4x-5)dx Answer:

Expand Binomials: Expand the integrand (x-1)(4x-5). To integrate the product of two binomials, we first need to expand the expression. (x-1)(4x-5) = 4x^2 - 5x - 4x + 5 = 4x^2 - 9x + 5Write Integral: Write the integral with the expanded integrand. Now we can write the integral as: ∫(4x^2 - 9x + 5)dxIntegrate Separately: Integrate each term separately. The integral of a sum is the sum of the integrals, so we can integrate each term separately. ∫4x^2dx - ∫9xdx + ∫5dxApply Power Rule: Apply the power rule for integration to each term. The power rule states that ∫x^n dx = (x^(n+1))/(n+1) + C, where C is the constant of integration. ∫4x^2dx = 4 * (x^(2+1))/(2+1) = (4/3)x^3 ∫9xdx = 9 * (x^(1+1))/(1+1) = (9/2)x^2 ∫5dx = 5xCombine Integrated Terms: Combine the integrated terms and add the constant of integration. Now we combine all the integrated terms and add the constant of integration C. (4/3)x^3 - (9/2)x^2 + 5x + C

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Evaluate the integral.

int(x-1)(4x-5)dx
Answer:

Evaluate the integral. $\newline$ $\int(x-1)(4 x-5) \mathrm{d} x$ $\newline$ Answer:

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Calculus

Find indefinite integrals using the substitution and by parts

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Q. Evaluate the integral.

\newline

\int(x-1)(4 x-5) \mathrm{d} x

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Answer:

Expand Binomials: Expand the integrand $(x-1)(4x-5)$ . To integrate the product of two binomials, we first need to expand the expression. $(x-1)(4x-5) = 4x^2 - 5x - 4x + 5 = 4x^2 - 9x + 5$
Write Integral: Write the integral with the expanded integrand. $\newline$ Now we can write the integral as: $\newline$ $\int(4x^2 - 9x + 5)\,dx$
Integrate Separately: Integrate each term separately. $\newline$ The integral of a sum is the sum of the integrals, so we can integrate each term separately. $\newline$ $\int 4x^2 \, dx - \int 9x \, dx + \int 5 \, dx$
Apply Power Rule: Apply the power rule for integration to each term. $\newline$ The power rule states that $\int x^n \, dx = \frac{x^{(n+1)}}{n+1} + C$ , where $C$ is the constant of integration. $\newline$ $\int 4x^2\,dx = 4 \cdot \frac{x^{(2+1)}}{2+1} = \frac{4}{3}x^3$ $\newline$ $\int 9x\,dx = 9 \cdot \frac{x^{(1+1)}}{1+1} = \frac{9}{2}x^2$ $\newline$ $\int 5\,dx = 5x$
Combine Integrated Terms: Combine the integrated terms and add the constant of integration. $\newline$ Now we combine all the integrated terms and add the constant of integration $C$ . $\newline$ $\frac{4}{3}x^3 - \frac{9}{2}x^2 + 5x + C$