int(x^(2)-3x+5)/(sqrtx)dx

Break into separate terms: Let's start by simplifying the integral by breaking it into separate terms that can be integrated individually. I = ∫(x^2 - 3x + 5) / sqrt(x) dx I = ∫x^(3/2) dx - 3∫x^(1/2) dx + 5∫x^(-1/2) dxIntegrate each term: Now we will integrate each term separately using the power rule for integration, which states that ∫x^n dx = x^(n+1)/(n+1) + C, where n ≠ -1. First term: ∫x^(3/2) dx = (2/5)x^(5/2) + C1 Second term: -3∫x^(1/2) dx = -3(2/3)x^(3/2) + C2 Third term: 5∫x^(-1/2) dx = 5(2)x^(1/2) + C3Simplify constants and combine: Now we will simplify the constants and combine the terms. First term: (2/5)x^(5/2) + C1 Second term: -2x^(3/2) + C2 Third term: 10x^(1/2) + C3Combine constants into single constant: Combine the constants C1, C2, and C3 into a single constant C, since the sum of constants is also a constant. I = (2/5)x^(5/2) - 2x^(3/2) + 10x^(1/2) + CFind indefinite integral: We have found the indefinite integral of the function. Final answer: I = (2/5)x^(5/2) - 2x^(3/2) + 10x^(1/2) + C

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

$\int \frac{x^{2}-3x+5}{\sqrt{x}} \, dx$

View step-by-step help

Home

Math Problems

Calculus

Find indefinite integrals using the substitution and by parts

Full solution

\int \frac{x^{2}-3x+5}{\sqrt{x}} \, dx

Break into separate terms: Let's start by simplifying the integral by breaking it into separate terms that can be integrated individually. $\newline$ $I = \int(\frac{x^2 - 3x + 5}{\sqrt{x}}) \, dx$ $\newline$ $I = \int x^{\frac{3}{2}} \, dx - 3\int x^{\frac{1}{2}} \, dx + 5\int x^{-\frac{1}{2}} \, dx$
Integrate each term: Now we will integrate each term separately using the power rule for integration, which states that $\int x^n \, dx = \frac{x^{n+1}}{n+1} + C$ , where $n \neq -1$ .
First term: $\int x^{\frac{3}{2}} \, dx = \frac{2}{5}x^{\frac{5}{2}} + C_1$
Second term: $-3\int x^{\frac{1}{2}} \, dx = -3\left(\frac{2}{3}x^{\frac{3}{2}}\right) + C_2$
Third term: $5\int x^{-\frac{1}{2}} \, dx = 5(2)x^{\frac{1}{2}} + C_3$
Simplify constants and combine: Now we will simplify the constants and combine the terms. $\newline$ First term: $(\frac{2}{5})x^{(\frac{5}{2})} + C_1$ $\newline$ Second term: $-2x^{(\frac{3}{2})} + C_2$ $\newline$ Third term: $10x^{(\frac{1}{2})} + C_3$
Combine constants into single constant: Combine the constants $C_1$ , $C_2$ , and $C_3$ into a single constant $C$ , since the sum of constants is also a constant. $\newline$ $I = \frac{2}{5}x^{\frac{5}{2}} - 2x^{\frac{3}{2}} + 10x^{\frac{1}{2}} + C$
Find indefinite integral: We have found the indefinite integral of the function. $\newline$ Final answer: $I = \frac{2}{5}x^{\frac{5}{2}} - 2x^{\frac{3}{2}} + 10x^{\frac{1}{2}} + C$