Verify the following integrals:
i. $\quad \int \ln u d u=u \ln u-u+C$
ii. $\int \frac{1}{x \ln x} d x=\ln |\ln x|+C$
iii. $\int t\left(e^{t}\right) d t=e^{t}(t-1)+C$
Answer
Thus, all three integral formulas are verified. The derivatives of the right hand side of the equations are: \(\boxed{\ln u}\), \(\boxed{\frac{1}{x \ln x}}\), and \(\boxed{t e^{t}}\).
Step 1 ：We are given three integral formulas to verify. We can do this by taking the derivative of the right hand side of each equation and checking if it equals to the integrand on the left hand side. This is because the derivative of the integral of a function is the function itself.
Step 2 ：For the first integral, \(\int \ln u d u\), the right hand side is \(u \ln u-u+C\). Taking the derivative of this gives us \(\ln u\), which matches the integrand.
Step 3 ：For the second integral, \(\int \frac{1}{x \ln x} d x\), the right hand side is \(\ln |\ln x|+C\). Taking the derivative of this gives us \(\frac{1}{x \ln x}\), which matches the integrand.
Step 4 ：For the third integral, \(\int t\left(e^{t}\right) d t\), the right hand side is \(e^{t}(t-1)+C\). Taking the derivative of this gives us \(t e^{t}\), which matches the integrand.
Step 5 ：Thus, all three integral formulas are verified. The derivatives of the right hand side of the equations are: \(\boxed{\ln u}\), \(\boxed{\frac{1}{x \ln x}}\), and \(\boxed{t e^{t}}\).
