Calculate the integral and write the answer in simplest form. int(2x^(2)-3x^(4)+5x^(-3))dx Answer:

Given Integral: We are given the integral of a polynomial function: int(2x^2 - 3x^4 + 5x^(-3))dx To solve this, we will integrate each term separately using the power rule for integration, which states that the integral of x^n with respect to x is (x^(n+1))/(n+1) for n ≠ -1.Integrating 2x^2: First, we integrate the term 2x^2: int(2x^2)dx = 2 * int(x^2)dx = 2 * (x^(2+1))/(2+1) = (2/3)x^3Integrating -3x^4: Next, we integrate the term -3x^4: int(-3x^4)dx = -3 * int(x^4)dx = -3 * (x^(4+1))/(4+1) = (-3/5)x^5Integrating 5x^(-3): Finally, we integrate the term 5x^(-3): int(5x^(-3))dx = 5 * int(x^(-3))dx = 5 * (x^(-3+1))/(-3+1) = 5 * (x^(-2))/(-2) = -5/2 * x^(-2)Combining Integrated Terms: Now, we combine all the integrated terms and add the constant of integration C: Integral = (2/3)x^3 - (3/5)x^5 - (5/2)x^(-2) + C

Resources

Testimonials

Plans

Resources

Testimonials

Plans

AI tutor

Welcome to Bytelearn!

Let’s check out your problem:

Calculate the integral and write the answer in simplest form.

int(2x^(2)-3x^(4)+5x^(-3))dx
Answer:

Calculate the integral and write the answer in simplest form. $\newline$ $\int\left(2 x^{2}-3 x^{4}+5 x^{-3}\right) d x$ $\newline$ Answer:

View step-by-step help

Home

Math Problems

Calculus

Find indefinite integrals using the substitution

Full solution

Q. Calculate the integral and write the answer in simplest form.

\newline

\int\left(2 x^{2}-3 x^{4}+5 x^{-3}\right) d x

\newline

Answer:

Given Integral: We are given the integral of a polynomial function: $\int(2x^2 - 3x^4 + 5x^{-3})dx$ To solve this, we will integrate each term separately using the power rule for integration, which states that the integral of $x^n$ with respect to $x$ is $(x^{(n+1)})/(n+1)$ for $n \neq -1$ .
Integrating $2x^2$ : First, we integrate the term $2x^2$ :
$\int(2x^2)dx = 2 \times \int(x^2)dx = 2 \times \frac{x^{2+1}}{2+1} = \frac{2}{3}x^3$
Integrating $-3x^4$ : Next, we integrate the term $-3x^4$ : $\int(-3x^4)dx = -3 \times \int(x^4)dx = -3 \times \frac{x^{4+1}}{4+1} = \left(-\frac{3}{5}\right)x^5$
Integrating $5x^{-3}$ : Finally, we integrate the term $5x^{-3}$ : $\int(5x^{-3})dx = 5 \times \int(x^{-3})dx = 5 \times \frac{x^{-3+1}}{-3+1} = 5 \times \frac{x^{-2}}{-2} = -\frac{5}{2} \times x^{-2}$
Combining Integrated Terms: Now, we combine all the integrated terms and add the constant of integration $C$ :
Integral = $\frac{2}{3}x^3 - \frac{3}{5}x^5 - \frac{5}{2}x^{-2} + C$