How do you integrate ∫x2√x2+1 by trigonometric substitution? Calculus Techniques of Integration Integration by Trigonometric Substitution 1 Answer Cem Sentin · Soumalya Pramanik May 26, 2018 ∫x2⋅dx√x2+1=x2⋅√x2+1−12sinh−1x+C Explanation: ∫x2⋅dx√x2+1 After using x=sinhy and dx=coshy⋅dy transforms, this integral became ∫(sinhy)2⋅coshy⋅dy√(sinhy)2+1 =∫(sinhy)2⋅coshy⋅dy√(coshy)2 =∫(sinhy)2⋅coshy⋅dycoshy =∫(sinhy)2⋅dy =∫cosh2y−12⋅dy =14sinh2y−y2+C =14⋅2sinhy⋅coshy−y2+C =12sinhy⋅coshy−y2+C After using x=sinhy, coshy=√x2+1 and y=sinh−1x inverse transforms, I found ∫x2⋅dx√x2+1=x2⋅√x2+1−12sinh−1x+C Answer link Related questions How do you find the integral ∫1x2⋅√x2−9dx ? How do you find the integral ∫x3√x2+9dx ? How do you find the integral ∫x3⋅√9−x2dx ? How do you find the integral ∫x3√16−x2dx ? How do you find the integral ∫√x2−1xdx ? How do you find the integral ∫√x2−9x3dx ? How do you find the integral ∫x√x2+x+1dx ? How do you find the integral ∫dt√t2−6t+13 ? How do you find the integral ∫x⋅√1−x4dx ? How do you prove the integral formula ∫dx√x2+a2=ln(x+√x2+a2)+C ? See all questions in Integration by Trigonometric Substitution Impact of this question 15407 views around the world You can reuse this answer Creative Commons License