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Question Number 22166 by Joel577 last updated on 12/Oct/17

$$\mathrm{If}\underset{1}{\stackrel{4}{\int }}f\left(x\right)dx=5$$$$\mathrm{what}\mathrm{is}\mathrm{the}\mathrm{value}\mathrm{of}\underset{0}{\stackrel{1}{\int }}f(3x+1)dx?$$

Answered by ajfour last updated on 12/Oct/17

$$I={\int }_0^{1}f(3x+1)dx$$$$let3x+1=t\Rightarrow dx=\frac{dt}{3}$$$$clearlythenx=0\Rightarrow t=1$$$$andwhenx=1,t=4$$$$soI=\frac{1}{3}{\int }_1^{4}f\left(t\right)dt=\frac{1}{3}{\int }_1^{4}f\left(\rho \right)d\rho$$$$=\frac{1}{3}{\int }_1^{4}f\left(x\right)dx=\frac{1}{3}\times 5=\frac{5}{3}.$$

Commented by Joel577 last updated on 13/Oct/17

$$thankyouverymuch$$

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