Solve 12 / 2 + c ∫ (from - 1 to 12) of int _ - 1^1x^2e^x^3 wrt x=e^2

$$\frac { 12 } { 2 + c } \int _ { - 1 } ^ { 12 } \int _ { - 1 } ^ { 1 } x ^ { 2 } e ^ { x ^ { 3 } } d x = e ^ { 2 }$$

Solve for c

$$c=-\frac{52}{e^{3}}+\frac{52}{e}-2\approx 14.540803386$$

Steps for Solving Linear Equation

Variable c cannot be equal to -2 since division by zero is not defined. Multiply both sides of the equation by c+2.

$$12\int _{-1}^{12}\int _{-1}^{1}x^{2}e^{x^{3}}\mathrm{d}x\mathrm{d}x=\left(c+2\right)e^{2}$$

Use the distributive property to multiply c+2 by e^{2}.

$$12\int _{-1}^{12}\int _{-1}^{1}x^{2}e^{x^{3}}\mathrm{d}x\mathrm{d}x=ce^{2}+2e^{2}$$

Swap sides so that all variable terms are on the left hand side.

$$ce^{2}+2e^{2}=12\int _{-1}^{12}\int _{-1}^{1}x^{2}e^{x^{3}}\mathrm{d}x\mathrm{d}x$$

Subtract 2e^{2} from both sides.

$$ce^{2}=12\int _{-1}^{12}\int _{-1}^{1}x^{2}e^{x^{3}}\mathrm{d}x\mathrm{d}x-2e^{2}$$

The equation is in standard form.

$$e^{2}c=-\frac{52}{e}-2e^{2}+52e$$

Divide both sides by e^{2}.

$$\frac{e^{2}c}{e^{2}}=\frac{-\frac{52}{e}-2e^{2}+52e}{e^{2}}$$

Dividing by e^{2} undoes the multiplication by e^{2}.

$$c=\frac{-\frac{52}{e}-2e^{2}+52e}{e^{2}}$$

Divide 52e-\frac{52}{e}-2e^{2} by e^{2}.

$$c=-\frac{52}{e^{3}}+\frac{52}{e}-2$$

Variable c cannot be equal to -2.

$$c=-\frac{52}{e^{3}}+\frac{52}{e}-2\text{, }c\neq -2$$