How do you integrate #int cosxsqrt(9+25sin^2x)# using trig substitutions?

Question

Christopher Agresta · Accepted Answer

#(5sinxsqrt(9+25sin^2x)+9lnabs(sqrt(9+25sin^2x)+5sinx))/10+C# #intcosxsqrt(9+25sin^2x)dx# First apply the non-trigonometric substitution #sinx=u# such that #cosxdx=du#. #=intsqrt(9+25u^2)du# Now, we will apply the trigonometric substitution #u=3/5tantheta#. This implies that #du=3/5sec^2thetad theta#. These yield: #=intsqrt(9+9tan^2theta)(3/5sec^2thetad theta)# #=9/5intsqrt(1+tan^2theta)(sec^2theta d theta)# Recall that #1+tan^2theta=sec^2theta#: #=9/5intsec^3thetad theta# The process of finding #intsec^3thetad theta# is fairly complex, so click here to see how it's done. #=9/10(secthetatantheta+lnabs(sectheta+tantheta))+C# Note that #tantheta=(5u)/3#. This means we have a right triangle with an opposite side of #5u# and an adjacent side of #3#, resulting in a hypotenuse of #sqrt(9+25u^2)#. The secant of this triangle is equal to the hypotenuse over the adjacent side, or #sqrt(9+25u^2)/3#. Replace these values in the solved integral to simplify: #=9/10(sqrt(9+25u^2)/3((5u)/3)+lnabs(sqrt(9+25u^2)/3+(5u)/3))+C# #=(5usqrt(9+25u^2))/10+9/10lnabs(1/3(sqrt(9+25u^2)+5u))+C# Note that the #1/3# in the logarithm can be taken out as a constant and be absorbed with #C#, the constant of integration. #=(5usqrt(9+25u^2)+9lnabs(sqrt(9+25u^2)+5u))/10+C# Since #u=sinx#: #=(5sinxsqrt(9+25sin^2x)+9lnabs(sqrt(9+25sin^2x)+5sinx))/10+C#

Christopher Agresta · Answer

undefined To integrate ∫cos(x)√(9 + 25sin^2(x)) using trigonometric substitution, follow these steps:

1. Recognize that the integrand involves a square root with a trigonometric function inside it.
2. Substitute sin(x) = 3/5tan(θ) to simplify the expression.
3. Use the Pythagorean identity sin^2(x) + cos^2(x) = 1 to express cos(x) in terms of sin(x).
4. Express the square root term in the integrand using the substitution sin(x) = 3/5tan(θ).
5. Substitute all the trigonometric functions in terms of θ.
6. Simplify the expression and integrate with respect to θ.
7. Finally, substitute back the expression in terms of x.

Following these steps will allow you to integrate the given expression using trigonometric substitution.