Show solutionHide solution+ Question 6 Evaluate the integral ∫xe1/(1+x2)(1+x2)2dx.\int \frac{x\,e^{1/(1+x^2)}}{(1+x^2)^2}\,dx. Question 6 - Solution Use substitution. Let u=11+x2.u=\frac{1}{1+x^2}. Then du=−2x(1+x2)2dx⇒−12du=x(1+x2)2dx.du=-\frac{2x}{(1+x^2)^2}\,dx \quad\Rightarrow\quad -\frac{1}{2}\,du=\frac{x}{(1+x^2)^2}\,dx. Substitute into the integral: ∫xe1/(1+x2)(1+x2)2dx=−12∫eudu.\int \frac{x\,e^{1/(1+x^2)}}{(1+x^2)^2}\,dx = -\frac{1}{2}\int e^{u}\,du. Integrate: −12∫eudu=−12eu.-\frac{1}{2}\int e^{u}\,du = -\frac{1}{2}e^{u}. Substitute back: −12e1/(1+x2)+C\boxed{ -\frac{1}{2}e^{1/(1+x^2)}+C }