Evaluate the following integrals. ∫56x(x−5)10dx

Question

Evaluate the following integrals.  ∫56x(x−5)10dx

John Timms · Accepted Answer

Step 1  Solution -   Given integral -   y=∫56x(x−5)10dx  Let,  t=x-5  differentiating on both sides w.r.t x,  dtdx=1  dt=dx  Now change in the limits as per t,  At x=5,  t=5-5=0,  At x=6,  t=6-5=1.  Substituting these values in the integral given, we get  y=∫01(t−5)⋅t10dt  Step 2  We can write it as,  y=∫01(t11−5t10)dt  Integrating w.r.t t,  y=[t(11+1)(11+1)−5t(10+1)(10+1)]01  y=[t1212−5t1111]01  y=112−511  y=−49132

Rex Gibbons · Accepted Answer

Step 1: If f(x) is a continuous function from a to b, and if F(x) is its integral, then:

\int_{a}^{b} f (x) d x = F (x) ∣_{a}^{b} = F (b) - F (a)

Step 2: In this case,

f (x) = x {(x - 5)}^{10}

. Find its integral.

\frac{{(x - 5)}^{12}}{12} + \frac{5 {(x - 5)}^{11}}{11} ∣_{5}^{6}

Step 3: Since

F (x) ∣_{a}^{b} = F (b) - F (a)

, expand the above into F(6)−F(5):

(\frac{{(6 - 5)}^{12}}{12} + \frac{5 {(6 - 5)}^{11}}{11}) - (\frac{{(5 - 5)}^{12}}{12} + \frac{5 {(5 - 5)}^{11}}{11})

Step 4: Simplify

\frac{71}{132}

Evaluate the following integrals. \int_{5}^{6}x(x-5)^{10}dx

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