Find the definite integral of the following using a suitable substitution:
1) (x^2)/sqroot(x^3-1)dx
2)xe^(x^2)dx
3)(ln(x))^(7/2)/(x)dx
To find the definite integral using a suitable substitution, we follow these steps:
1) Identify a suitable substitution.
2) Substitute the given expression with the new variable.
3) Express the derivative (dx) in terms of the new variable.
4) Replace the integral limits with the new variable limits if necessary.
5) Integrate the expression.
6) Substitute the new variable back to the original variable if required.
Now, let's find the definite integrals for each problem:
1) ∫ (x^2)/√(x^3 - 1) dx
Substitution:
Let u = x^3 - 1
Differentiating both sides:
du/dx = 3x^2
=> dx = du/(3x^2)
Replacing the expression and the differential:
∫ (x^2)/√u * du/(3x^2)
Simplifying:
1/3 ∫ du/√u
Integrating:
1/3 * 2√u + C
= 2/3 √u + C
Substituting back the original variable:
= 2/3 √(x^3 - 1) + C
2) ∫ xe^(x^2) dx
Substitution:
Let u = x^2
Differentiating both sides:
du/dx = 2x
=> dx = du/(2x)
Replacing the expression and the differential:
∫ u * e^u * du/(2x)
Simplifying:
1/2 * ∫ e^u du
Integrating:
1/2 * e^u + C
= 1/2 * e^(x^2) + C
3) ∫ (ln(x))^(7/2)/x dx
Substitution:
Let u = ln(x)
Differentiating both sides:
du/dx = 1/x
=> dx = x du
Replacing the expression and the differential:
∫ u^(7/2) x du/x
Simplifying:
∫ u^(7/2) du
Integrating:
(2/9) * u^(9/2) + C
= (2/9) * (ln(x))^(9/2) + C
These are the solutions for the definite integrals using a suitable substitution method.