Evaluate the integral int_(Gamma)(e^(2))/(2*(2^(2)+1))dz

Pull Constants Out: We have the integral int(e^(2))/(2*(2^(2)+1))dz. Since e^2 is a constant and 2^2 + 1 is also a constant, we can pull them out of the integral. Calculation: int(e^(2))/(2*(2^(2)+1))dz = (e^(2))/(2*(2^(2)+1)) * int(1)dzIntegrate 1: Now we integrate 1 with respect to z. Calculation: int(1)dz = zSubstitute Integral of 1: Substitute the integral of 1 back into the equation. Calculation: (e^(2))/(2*(2^(2)+1)) * z + CSimplify Constants: Simplify the constants. Calculation: (e^(2))/(2*(4+1)) * z + C = (e^(2))/(2*5) * z + C = (e^(2))/(10) * z + C

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Math Problems

Calculus

Find indefinite integrals using the substitution

Full solution

Q. Evaluate the integral

\int_{\Gamma}\left(\frac{e^{2}}{2\cdot(2^{2}+1)}\right)dz

Pull Constants Out: We have the integral $\int \frac{e^{2}}{2(2^{2}+1)}dz$ . Since $e^2$ is a constant and $2^2 + 1$ is also a constant, we can pull them out of the integral. $\newline$ Calculation: $\int \frac{e^{2}}{2(2^{2}+1)}dz = \frac{e^{2}}{2(2^{2}+1)} \times \int 1dz$
Integrate $1$ : Now we integrate $1$ with respect to $z$ . $\newline$ Calculation: $\int(1)\,dz = z$
Substitute Integral of $1$ : Substitute the integral of $1$ back into the equation. $\newline$ Calculation: $\left(\frac{e^{2}}{2\cdot(2^{2}+1)}\right) \cdot z + C$
Simplify Constants: Simplify the constants. $\newline$ Calculation: (e^{$2$})/(\(2*( $4$ + $1$ )) * z + C = (e^{ $2$ })/( $2$ * $5$ ) * z + C = (e^{ $2$ })/( $10$ ) * z + C