Evaluate #int \ (1+sqrt(x))^9/sqrt(x) \ dx #?

Answer 1

#int (sqrt(x)+1)^9/(sqrt(x)) "d"x = (sqrt(x)+1)^10/5 + C#

Could use the binomial theorem to expand #(1+sqrt(x))^9# and integrate simply from there but this seems tedious.

If we consider the integral,

#int "f"'(x)["f"(x)]^(n) "d"x#.
Substitute #u="f"(x)#. Then #"d"x = 1/("f"'(x)) "d"u#,
#int ("f"'(x)u^n)/("f"'(x)) "d"u# = (u*(n+1))/(n+1) + C#.

We conclude that,

#int "f"'(x)["f"(x)]^(n) "d"x = ["f"(x)]^(n+1)/(n+1) + C#.
This is a very common and very useful technique for solving integrals. Notice that #"d"/("d"x) sqrt(x) = 1/(2sqrt(x))#.

We can therefore rewrite the given integral easily so it is in this form.

#int (sqrt(x)+1)^9/(sqrt(x)) "d"x = 2 int 1/(2sqrt(x)) (sqrt(x)+1)^9 "d"x#.

Using this general result we conclude,

#int (sqrt(x)+1)^9/(sqrt(x)) "d"x = 2( (sqrt(x)+1)^10/10 ) + C# #int (sqrt(x)+1)^9/(sqrt(x)) "d"x = (sqrt(x)+1)^10/5 + C#
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 2

# int \ (1+sqrt(x))^9/sqrt(x) \ dx = (1+sqrt(x))^10/5 + C #

We want to evaluate:

# I = int \ (1+sqrt(x))^9/sqrt(x) \ dx #

We can perform a simple substitution. Let:

# u = 1+sqrt(x) => (du)/dx = 1/(2sqrt(x)) #

Substituting into the integral we get:

# I = int \ u^9 \ 2 \ du # # \ \ = 2 \ int \ u^9 \ du # # \ \ = 2 u^10/10 + C # # \ \ = u^10/5 + C #

Restoring the substitution we get:

# I = (1+sqrt(x))^10/5 + C #
Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer 3

To evaluate ( \int \frac{(1+\sqrt{x})^9}{\sqrt{x}} , dx ), we can simplify the expression by expanding ( (1+\sqrt{x})^9 ) using the binomial theorem. After expanding, we integrate each term separately. The integral of ( \sqrt{x} ) is ( \frac{2}{3}x^{3/2} + C ).

Sign up to view the whole answer

By signing up, you agree to our Terms of Service and Privacy Policy

Sign up with email
Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

Not the question you need?

Drag image here or click to upload

Or press Ctrl + V to paste
Answer Background
HIX Tutor
Solve ANY homework problem with a smart AI
  • 98% accuracy study help
  • Covers math, physics, chemistry, biology, and more
  • Step-by-step, in-depth guides
  • Readily available 24/7