To determine
To evaluate:
The given definite integral ∫12xx-1 dx
Answer
1615
Explanation
1) Concept:
i) The substitution rule for definite integral:
If g'(x) is a continuous function on a,b whose f is continuous on range of u=g(x), then ∫abfgxg'(x)dx=∫g(a)g(b)f(u)du. Here,g(x) is substituted as u and g’(x)dx =du
ii) Indefinite integral
∫xn dx=xn+1n+1+C (n≠-1)
iii)
∫ab[fx+gx] dx=∫abfxdx+∫abgxdx
2) Given:
The given integral is
∫12xx-1 dx.
3) Calculations:
Here, use the substitution method.
Substitute x-1=u, then x=u+1.
Differentiating with respect to x
dx=du
The limits change and the new limits of integration are calculated by substituting
for x=1, u=1-1=0 & for x=2, u=2-1 =1
Therefore, the given integral becomes
∫12xx-1 dx=∫01(u+1)u du
=∫01u+1u1/2 du
=∫01u3/2+u1/2 du
=∫01u3/2 du+∫01u1/2 du
Integrating
=u32+132+101+u12+112+101
Simplifying this, we get
=u5/25/201+u3/23/201
Substituting the limits.
=25152-052+23132-032
Solving this, we get
=25(1-0)+23(1-0)
=25+23
=1615
Conclusion:
Therefore,
∫12xx-1 dx=1615