Show solutionHide solution+ Question 4 Evaluate the integral ∫xsin(x2)1+cos(x2)dx.\int \frac{x\,\sin(x^2)}{\sqrt{1+\cos(x^2)}}\,dx. Question 4 - Solution Use substitution. Let u=cos(x2).u = \cos(x^2). Then du=−sin(x2)⋅2xdx⇒−12du=xsin(x2)dx.du = -\sin(x^2)\cdot 2x\,dx \quad\Rightarrow\quad -\,\frac{1}{2}\,du = x\sin(x^2)\,dx. Substitute into the integral: ∫xsin(x2)1+cos(x2)dx=−12∫du1+u.\int \frac{x\,\sin(x^2)}{\sqrt{1+\cos(x^2)}}\,dx = -\frac{1}{2}\int \frac{du}{\sqrt{1+u}}. Integrate: −12∫(1+u)−1/2du=−12⋅2(1+u)1/2=−1+u.-\frac{1}{2}\int (1+u)^{-1/2}\,du = -\frac{1}{2}\cdot 2(1+u)^{1/2} = -\sqrt{1+u}. Substitute back: −1+cos(x2)+C\boxed{ -\sqrt{1+\cos(x^2)} + C }