guess and fudge/guess and check method\n8. use guess and fudge to find the antiderivatives:\n(i) $\\int…

guess and fudge/guess and check method\n8. use guess and fudge to find the antiderivatives:\n(i) $\\int \\cos (3 \\pi x) d x$\n(v) $\\int 4 x^{2}\\left(x^{3}+3\\right)^{7} d x$\n(ii) $\\int(3 x+1)^{5} d x$\n(vi) $\\int x \\sec ^{2}\\left(x^{2}+1\\right) d x$\n(iii) $\\int \\frac{1}{(3 x+5)^{2}} d x$\n(vii) $\\int \\frac{2 x}{\\left(x^{2}+1\\right)} d x$\n(iv) $\\int 2 x e^{-x^{2}} d x$\n

guess and fudge/guess and check method\n8. use guess and fudge to find the antiderivatives:\n(i) $\\int \\cos (3 \\pi x) d x$\n(v) $\\int 4 x^{2}\\left(x^{3}+3\\right)^{7} d x$\n(ii) $\\int(3 x+1)^{5} d x$\n(vi) $\\int x \\sec ^{2}\\left(x^{2}+1\\right) d x$\n(iii) $\\int \\frac{1}{(3 x+5)^{2}} d x$\n(vii) $\\int \\frac{2 x}{\\left(x^{2}+1\\right)} d x$\n(iv) $\\int 2 x e^{-x^{2}} d x$\n

Answer

Explanation:

Step1: Guess the antiderivative form

For (\int\cos(3\pi x)dx), we know that the antiderivative of (\cos(u)) is (\sin(u)). Guess (F(x)=\sin(3\pi x)).

Step2: Differentiate the guess

Differentiate (F(x)=\sin(3\pi x)) using the chain rule. (F^\prime(x)=\cos(3\pi x)\cdot3\pi).

Step3: Fudge to match the integrand

Since our guess differentiated gives (3\pi\cos(3\pi x)) and the integrand is (\cos(3\pi x)), we divide our guess by (3\pi). So the antiderivative is (\frac{1}{3\pi}\sin(3\pi x)+C).

Answer:

(\frac{1}{3\pi}\sin(3\pi x)+C)