Evaluate the integral. int(8x^(3)+7x^(2)-1)/(x^(2))dx Answer:

Simplify integrand: Simplify the integrand by dividing each term by x^2. We have the integral: int((8x^3 + 7x^2 - 1)/x^2)dx This simplifies to: int(8x + 7 - x^(-2))dxIntegrate each term: Integrate each term separately. The integral of 8x with respect to x is 8x^2/2. The integral of 7 with respect to x is 7x. The integral of x^(-2) with respect to x is -x^(-1). So we have: (8/2)x^2 + 7x - (-1)x^(-1) + CSimplify result: Simplify the result from Step 2. Simplifying the expression, we get: 4x^2 + 7x + x^(-1) + C This is the indefinite integral of the given function.

Home

Math Problems

Calculus

Find indefinite integrals using the substitution

Full solution

Q. Evaluate the integral.

\newline

\int \frac{8 x^{3}+7 x^{2}-1}{x^{2}} \mathrm{~d} x

\newline

Answer:

Simplify integrand: Simplify the integrand by dividing each term by $x^2$ . We have the integral: $\int\left(\frac{8x^3 + 7x^2 - 1}{x^2}\right)dx$ This simplifies to: $\int(8x + 7 - x^{-2})dx$
Integrate each term: Integrate each term separately. $\newline$ The integral of $8x$ with respect to $x$ is $\frac{8x^2}{2}$ . $\newline$ The integral of $7$ with respect to $x$ is $7x$ . $\newline$ The integral of $x^{-2}$ with respect to $x$ is $-x^{-1}$ . $\newline$ So we have: $\newline$ $(\frac{8}{2})x^2 + 7x - (-1)x^{-1} + C$
Simplify result: Simplify the result from Step $2$ . $\newline$ Simplifying the expression, we get: $\newline$ $4x^2 + 7x + x^{-1} + C$ $\newline$ This is the indefinite integral of the given function.