Question

$\int e^{x^4}(x+x^3+2x^5)e^{x^2}dx$ is equal to

• A. $\frac{1}{2}x{e}^{x^2}\cdot e^{x^4}+c$
• B. $\frac{1}{2}{x}^2{e}^{x^4}+c$
• C. $\frac{1}{2}{e}^{x^2}\cdot e^{x^4}+c$
• D. $\frac{1}{2}{x}^2{e}^{x^2}\cdot e^{x^4}+c$

Answer 1

The correct option is D $\frac{1}{2}{x}^2{e}^{x^2}\cdot e^{x^4}+c$

$f\left(x\right)=\int e^{x^2}{e}^{x^4}(x+x^3+2x^5)dxf\left(x\right)=\int e^{x^2}{e}^{x^4}2x^{5}dx+\int e^{x^2}{e}^{x^4}(x+x^3)dxf\left(x\right)=\frac{1}{2}\int e^{x^2}{e}^{x^4}4x^{5}dx+\frac{1}{2}\int e^{x^2}{e}^{x^4}(2x+2x^3)dxf\left(x\right)=\frac{1}{2}\int \left(x^2{e}^{x^2}4x^3{e}^{x^4}\right)+e^{x^4}[e^{x^2}2x+{x^{2}e}^{x^2}2x]dxf\left(x\right)=\frac{1}{2}\int {d(x}^2{e}^{x^2}{e}^{x^4})f\left(x\right)=\frac{1}{2}{x}^2{e}^{x^2}{e}^{x^4}+c$