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Direct Integration, i.e., Integration without using 'u' substitution [Solved!]

My question

How do we directly integrate a function that involves an inverse trig function without using 'u' substitution?

Relevant page

1. Integration: The General Power Formula

What I've done so far

We integrated Examples 1, 2, and 4 without using 'u' substitution.

For instance, for example one, we added one to the exponent of sin x to obtain the new exponent of `4/3`. Then divided by `4/3`. The result is `3/4 sin^(4/3)x` + C.

Not as easy with example 3. Having trouble undoing an inverse trig function.

X

How do we directly integrate a function that involves an inverse trig function without using 'u' substitution?
Relevant page

<a href="/methods-integration/1-integration-power-formula.php">1. Integration: The General Power Formula</a>

What I've done so far

We integrated Examples 1, 2, and 4 without using 'u' substitution. 

For instance, for example one, we added one to the exponent of sin x to obtain the new exponent of `4/3`. Then divided by `4/3`.  The result is `3/4 sin^(4/3)x` + C.


Not as easy with example 3.  Having trouble undoing an inverse trig function.

Re: Direct Integration, i.e., Integration without using 'u' substitution

@Phinah. The major problems with your approach are:

(1) Where did the `cos x` go?

That is, it seems you are doing:

`int sin^(1/3)x dx = 3/4sin^(4/3)x + C`

But the left hand side of that equation is not what the question was asking. It actually has:

`int sin^(1/3)x cos x dx`

So in your working, how come `cos x` disappeared in the question, and how did you cater for it in the answer?

(2) We always know our integration is correct if we differentiate our answer. We should get back to what the question had.

Now `d/dx (3/4sin^(4/3)x + C) = sin^(1/3)x cos x`

Now, that right hand side is what Question (1) actually asks us to integrate, however, your question had

`sin^(1/3)x`,

without the `cos x`.

Does that make sense?

X

@Phinah. The major problems with your approach are:

(1) Where did the `cos x` go?

That is, it seems you are doing:

`int sin^(1/3)x dx = 3/4sin^(4/3)x + C`

But the left hand side of that equation is not what the question was asking. It actually has:

`int sin^(1/3)x cos x dx`

So in your working, how come `cos x` disappeared in the question, and how did you cater for it in the answer?

(2) We always know our integration is correct if we differentiate our answer. We should get back to what the question had.

Now `d/dx (3/4sin^(4/3)x + C) = sin^(1/3)x cos x`

Now, that right hand side is what Question (1) actually asks us to integrate, however, your question had 

`sin^(1/3)x`, 

without the `cos x`.

Does that make sense?

Re: Direct Integration, i.e., Integration without using 'u' substitution

Right, after we differentiated our solutions, we arrived at the original integrands. Which is why we thought it was okay to not include the derivative of what we consider 'u' or the focal point of the integrand.

X

Right, after we differentiated our solutions, we arrived at the original integrands.  Which is why we thought it was okay to not include the derivative of what we consider 'u' or the focal point of the integrand.

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