For the sake of not being ambiguous, please use parentheses to clarify what is and isn't under the radical.
a.) Acceptable trigonometric functions to use here would be cosine, sine, tanh, or sech, since those are the trig functions such that 1-f(x)² is the square of another trig function.
∫√(1-100x²) dx
x = sin(θ)/10, dx = cos(θ)/10 dθ
∫ √(1-sin²(θ))*cos(θ)/10 dθ
∫ cos²(θ)/10 dθ
(θ+cos(θ)sin(θ))/20 + C
(sin(-1)(10x)+10x√(1-100x²))/20 + C
b.) Acceptable trigonometric functions to use here would be secant, cosecant, cosh, or coth, since those are the trig functions such that f(x)²-1 is the square of another trig function.
∫√(25x²-36)dx
x = 6sec(θ)/5, dx = 6sec(θ)tan(θ)/5 dθ
∫ √(36sec²(θ)-36) * 6sec(θ)tan(θ)/5 dθ
∫ 6tan(θ) * 6sec(θ)tan(θ)/5 dθ
Integration by parts:
6sec(θ)/5 * 6tan(θ) - ∫ 6sec(θ)/5 * 6sec(θ)² dθ
36sec(θ)tan(θ)/5 - ∫ 36sec(θ)(tan(θ)²+1)/5 dθ
36sec(θ)tan(θ)/5 - 36ln(sec(θ)+tan(θ))/5 - ∫ 36sec(θ)tan(θ)²/5 dθ
∫ 6tan(θ) * 6sec(θ)tan(θ)/5 dθ = 36sec(θ)tan(θ)/5 - 36ln(sec(θ)+tan(θ))/5 - ∫ 36sec(θ)tan(θ)²/5 dθ
∫ 36sec(θ)tan(θ)²/5 dθ = 36sec(θ)tan(θ)/5 - 36ln(sec(θ)+tan(θ))/5 - ∫ 36sec(θ)tan(θ)²/5 dθ
∫ 72sec(θ)tan(θ)²/5 dθ = 36sec(θ)tan(θ)/5 - 36ln(sec(θ)+tan(θ))/5
∫ 36sec(θ)tan(θ)²/5 dθ = 18sec(θ)tan(θ)/5 - 18ln(sec(θ)+tan(θ))/5
18(sec(θ)tan(θ) - ln(sec(θ)+tan(θ)))/5 + C
18(5x/6*√(25x²-36)/6 - ln(5x/6+√(25x²-36)/6))/5 + C
x√(25x²-36)/2 - 18ln(5x+√(25x²-36))/5 + C
(This is also the same as x√(25x²-36)/2 - 18acosh(5x/6)/5 + C)
c.) Acceptable trigonometric functions to use here would be tangent, cotangent, sinh, and csch, since those are functions such that f(x)²+1 is the square of another trig function.
∫ √(16x²+36) dx
x = 6tan(θ)/4, dx = 6sec(θ)²/4 dθ
∫ √(36tan(θ)²+36) * 6sec(θ)²/4 dθ
∫ 9sec(θ)³ dθ
∫ 9sec(θ)(tan(θ)²+1) dθ
In b), we already determined that
∫ 36sec(θ)tan(θ)²/5 dθ = 18(sec(θ)tan(θ) - ln(sec(θ)+tan(θ)))/5 + C
∫ sec(θ)tan(θ)² dθ = (sec(θ)tan(θ) - ln(sec(θ)+tan(θ)))/2 + C
9ln(sec(θ)+tan(θ)) + 9(sec(θ)tan(θ) - ln(sec(θ)+tan(θ)))/2 + C
9(sec(θ)tan(θ) + ln(sec(θ)+tan(θ)))/2 + C
9(4x/6*√(16x²+36)/6 + ln(4x/6+√(16x²+36)/6))/2 + C
x√(4x²+9) + 9ln(2x+√(4x²+9))/2 + C
(This is also the same as x√(4x²+9) + 9asinh(2x/3)/2 + C)
