Evaluate the indefinite integral
(a.)∫√(cotx)csc^2xdx
(b.)∫sec^3xtanxdx
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(a.) To evaluate the indefinite integral of √(cot(x))csc^2(x)dx, we can use the substitution method.
Let's make the substitution u = cot(x). Then du = -csc^2(x)dx.
Now, we need to express the other parts of the integral in terms of u:
- cot(x) = u
- csc^2(x)dx = -du
So, the integral becomes ∫√(u)(-du). We can rewrite this as -∫√(u)du.
To simplify further, we can rewrite √(u) as u^(1/2). Now, the integral becomes -∫u^(1/2)du.
Using the power rule for integration, we add 1 to the exponent and divide by the new exponent:
-∫u^(1/2)du = -(2/3)u^(3/2) + C,
where C represents the constant of integration.
Finally, substituting u back in terms of x, we have -(2/3)cot^(3/2)(x) + C as the solution to the indefinite integral ∫√(cot(x))csc^2(x)dx.
(b.) To evaluate the indefinite integral of sec^3(x)tan(x)dx, we can use integration by parts.
Integration by parts formula:
∫u(x)v'(x)dx = u(x)v(x) - ∫v(x)u'(x)dx
In this case, let u = sec^2(x) and dv = sec(x)tan(x)dx.
Now, let's find the derivatives of u and the antiderivative of dv:
- Differentiating u, we get du = 2sec(x)tan(x)dx
- Integrating dv, we get v = ∫sec(x)tan(x)dx. This integral is a common integral with the result of ln|sec(x) + tan(x)|.
Using the integration by parts formula, we have:
∫sec^3(x)tan(x)dx = sec^2(x) ln|sec(x) + tan(x)| - ∫2sec(x)tan(x) * sec^2(x)dx.
Simplifying further, we have:
∫sec^3(x)tan(x)dx = sec^2(x) ln|sec(x) + tan(x)| - 2∫sec^3(x)tan(x)dx.
Now, let's move the integral ∫sec^3(x)tan(x)dx to the other side:
∫sec^3(x)tan(x)dx + 2∫sec^3(x)tan(x)dx = sec^2(x) ln|sec(x) + tan(x)|.
Combining like terms, we obtain:
3∫sec^3(x)tan(x)dx = sec^2(x) ln|sec(x) + tan(x)|.
Finally, dividing both sides by 3 gives us the solution to the indefinite integral:
∫sec^3(x)tan(x)dx = (1/3)sec^2(x) ln|sec(x) + tan(x)| + C,
where C represents the constant of integration.