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 solve each of these integrals using substitution, we need to identify a suitable substitution that simplifies the integrand. Let's go through each problem step by step.
1) ∫ (x^2)/sqrt(x^3-1) dx
Let's set u = x^3 - 1.
Now, differentiate both sides with respect to x: du/dx = 3x^2.
Solving for dx, we get dx = du/(3x^2).
Substituting dx and u into the integral, we have:
∫ [(x^2)/sqrt(u)] * (du/(3x^2)).
Simplifying the expression, we can cancel out the x^2 terms:
∫ [1/sqrt(u)] * (du/3).
Now, let's integrate:
(1/3) ∫ (1/sqrt(u)) du.
This is a straightforward integral:
(1/3) * 2sqrt(u) + C.
To obtain the final answer in terms of x, substitute the original value of u:
(1/3) * 2sqrt(x^3 - 1) + C.
2) ∫ xe^(x^2) dx
Let's set u = x^2.
Now, differentiate both sides with respect to x: du/dx = 2x.
Solving for dx, we get dx = du/(2x).
Substituting dx and u into the integral, we have:
∫ x * e^u * (du/(2x)).
Simplifying the expression, we can cancel out the x terms:
(1/2) ∫ e^u du.
Integrating e^u gives us:
(1/2) * e^u + C.
Now, substitute the original value of u:
(1/2) * e^(x^2) + C.
3) ∫ (ln(x))^(7/2) / x dx
Let's set u = ln(x).
Now, differentiate both sides with respect to x: du/dx = 1/x.
Solving for dx, we get dx = x du.
Substituting dx and u into the integral, we have:
∫ u^(7/2) x du.
Now, we can separate the x and u terms:
∫ x * u^(7/2) du.
Integrating u^(7/2) gives us:
(2/9) * u^(9/2) + C.
Finally, substitute the original value of u:
(2/9) * (ln(x))^(9/2) + C.
Remember to always check your result by differentiating it to verify that it matches the original integrand.