How do you integrate #int x/sqrt(3 + x^2)dx# using trigonometric substitution? Calculus Techniques of Integration Integration by Trigonometric Substitution 1 Answer Cem Sentin Mar 28, 2018 #int x/sqrt(3+x^2)*dx=sqrt(x^2+3)+C# Explanation: #int x/sqrt(3+x^2)*dx# After using #x=sqrt3tany# and #dx=sqrt3*(secy)^2*dy# transforms, this integral became #int sqrt3tany*(sqrt3*(secy)^2*dy)/(sqrt3*secy)# =#int sqrt3secy*tany*dy# =#sqrt3secy+C# After using #tany=x/sqrt3# and #secy=sqrt(x^2+3)/sqrt3# inverse transforms, I found #int x/sqrt(3+x^2)*dx=sqrt3*sqrt(x^2+3)/sqrt3+C=sqrt(x^2+3)+C# Answer link Related questions How do you find the integral #int1/(x^2*sqrt(x^2-9))dx# ? How do you find the integral #intx^3/(sqrt(x^2+9))dx# ? How do you find the integral #intx^3*sqrt(9-x^2)dx# ? How do you find the integral #intx^3/(sqrt(16-x^2))dx# ? How do you find the integral #intsqrt(x^2-1)/xdx# ? How do you find the integral #intsqrt(x^2-9)/x^3dx# ? How do you find the integral #intx/(sqrt(x^2+x+1))dx# ? How do you find the integral #intdt/(sqrt(t^2-6t+13))# ? How do you find the integral #intx*sqrt(1-x^4)dx# ? How do you prove the integral formula #intdx/(sqrt(x^2+a^2)) = ln(x+sqrt(x^2+a^2))+ C# ? See all questions in Integration by Trigonometric Substitution Impact of this question 5218 views around the world You can reuse this answer Creative Commons License