Calculate the integral and write the answer in simplest form. int(5x+1)dx Answer:

Recognize Integral as Sum: Recognize the integral as the sum of two simpler integrals. The integral of a sum is the sum of the integrals, so we can write: ∫(5x+1)dx = ∫5xdx + ∫1dxIntegrate Each Term: Integrate each term separately. The integral of 5x with respect to x is (5/2)x^2, because the antiderivative of x^n is (x^(n+1))/(n+1) for n ≠ -1. The integral of 1 with respect to x is x, because the antiderivative of a constant is the constant times the variable. So we have: ∫5xdx = (5/2)x^2 ∫1dx = xCombine Results and Add Constant: Combine the results of the two integrals and add the constant of integration. The combined result is: (5/2)x^2 + x + C where C is the constant of integration.

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Find indefinite integrals using the substitution and by parts

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

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\int(5 x+1) d x

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Answer:

Recognize Integral as Sum: Recognize the integral as the sum of two simpler integrals. The integral of a sum is the sum of the integrals, so we can write: \int(\(5x+ $1$ )\,dx = \int $5$ x\,dx + \int $1$ \,dx
Integrate Each Term: Integrate each term separately. $\newline$ The integral of $5x$ with respect to $x$ is $(5/2)x^2$ , because the antiderivative of $x^n$ is $(x^{(n+1)})/(n+1)$ for $n \neq -1$ . $\newline$ The integral of $1$ with respect to $x$ is $x$ , because the antiderivative of a constant is the constant times the variable. $\newline$ So we have: $\newline$ $\int 5x\,dx = (5/2)x^2$ $\newline$ $x$ $0$
Combine Results and Add Constant: Combine the results of the two integrals and add the constant of integration. The combined result is: $(\frac{5}{2})x^2 + x + C$ where $C$ is the constant of integration.