1. Iff and g are differentiable functions, then the Product Rule for derivatives states that (f(x)g(x))...

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1. Iff and g are differentiable functions, then the Product Rule for derivatives states that (f(x)g(x)) = By integrating BOTH sides of the above equation, we get [[f(x)g'(x) + g(x)f'(x)] dx = f(x)g(x) (For now, we omit the customary "+C".) We can split the left-hand side into the sum of two integrals and subtract one of them from the right-hand side to get the integration by parts formula (see page 472): ff(x)g'(x) dx = -S. (The customary"+C" is now handled by the integral on the right-hand side.) This formula can sometimes be used to reduce a difficult integral (the left-hand side) to a potentially simpler integral (on the right-hand side). We can make this formula more convenient to use by converting to differential notation: let u = f(x) and v= g(x), so that du = f'(x) dx and dv= g'(x) dx. Then, the integration by parts formula above becomes (see page 472): fudv= -S- 2. One of the most difficult aspects of using integration by parts is knowing what to choose for u and dv. The following rule of thumb can be used to help with this: choose u to be whichever type of function comes first in the following list. L: Logarithmic functions (In(x), log, (x), etc.) I: Inverse trigonometric functions (tan-¹(x), arcsin(5x), etc.) A: Algebraic functions (x²,x+1, etc. T: Trigonometric functions (tan(x), cos(x²), etc.) E: Exponential functions (e*, 32x, etc.) dv is then chosen as the remaining part of the integrand. Use LIATE to identify a good choice for u and dv for the functions below: a) x5 In(x): u = b) x sin(x) cos(x): u= dv= dy = dx dx 1. Iff and g are differentiable functions, then the Product Rule for derivatives states that (f(x)g(x)) = By integrating BOTH sides of the above equation, we get [[f(x)g'(x) + g(x)f'(x)] dx = f(x)g(x) (For now, we omit the customary "+C".) We can split the left-hand side into the sum of two integrals and subtract one of them from the right-hand side to get the integration by parts formula (see page 472): ff(x)g'(x) dx = -S. (The customary"+C" is now handled by the integral on the right-hand side.) This formula can sometimes be used to reduce a difficult integral (the left-hand side) to a potentially simpler integral (on the right-hand side). We can make this formula more convenient to use by converting to differential notation: let u = f(x) and v= g(x), so that du = f'(x) dx and dv= g'(x) dx. Then, the integration by parts formula above becomes (see page 472): fudv= -S- 2. One of the most difficult aspects of using integration by parts is knowing what to choose for u and dv. The following rule of thumb can be used to help with this: choose u to be whichever type of function comes first in the following list. L: Logarithmic functions (In(x), log, (x), etc.) I: Inverse trigonometric functions (tan-¹(x), arcsin(5x), etc.) A: Algebraic functions (x²,x+1, etc. T: Trigonometric functions (tan(x), cos(x²), etc.) E: Exponential functions (e*, 32x, etc.) dv is then chosen as the remaining part of the integrand. Use LIATE to identify a good choice for u and dv for the functions below: a) x5 In(x): u = b) x sin(x) cos(x): u= dv= dy = dx dx

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Solution and g d fx gx dr a b are Thus integrating on both sids n gx gx fx dx ... View the full answer