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int(4x^(2)-6)(5x^(2)+3)dx=dots

$\int\left(4 x^{2}-6\right)\left(5 x^{2}+3\right) d x=\ldots$

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Evaluate definite integrals using the power rule

Full solution

Q.

\int\left(4 x^{2}-6\right)\left(5 x^{2}+3\right) d x=\ldots

Expand and Simplify: We are given the integral of the product of two polynomials, which we can expand before integrating: $\newline$ $\int (4x^2 - 6)(5x^2 + 3) \, dx$ $\newline$ First, we'll expand the product of the polynomials: $\newline$ $(4x^2 - 6)(5x^2 + 3) = 20x^4 + 12x^2 - 30x^2 - 18$ $\newline$ Simplify the middle terms: $\newline$ $20x^4 - 18x^2 - 18$ $\newline$ Now we can integrate term by term.
Integrate First Term: Integrate the first term: $\newline$ $\int 20x^4 \, dx = \frac{20}{5}x^5 = 4x^5$
Integrate Second Term: Integrate the second term: $\newline$ $\int -18x^2 \, dx = \frac{-18}{3}x^3 = -6x^3$
Integrate Constant Term: Integrate the constant term: $\newline$ $\int -18 \, dx = -18x$
Combine Integrated Terms: Combine the integrated terms to get the indefinite integral: $\newline$ $\int (4x^2 - 6)(5x^2 + 3) \, dx = 4x^5 - 6x^3 - 18x + C$ $\newline$ where $C$ is the constant of integration.

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