# Solve Integration: x^5cos(x^3) dx

• mckallin
In summary, The conversation is about solving the integration \int x^5cos(x^3) dx. The person asking for help has tried using the substitution method and by parts, but is still unable to solve the problem. They are looking for more advice and are considering using partial substitution.
mckallin

can anyone help me solve the following integration? thanks a lot.

$$\int$$ $$x^5$$cos$$(x^3)$$ dx

you really should show some more effort. ie. show us what you have tried, what you know and tell us what you have problems with.hint: try look for a substitution to make the integrand look nicer, then you can try by parts.

I have tried to do it by part.

If I make $$u=cos(x^3), dv=x^5 dx$$, the grade of x, which is in the $$cos(x^3)$$, won't be reduce.

If I make $$u=x^5, dv=cos(x^3) dx$$, I can't solve the $$\int cos(x^3) dx$$.

I have thought if there is some way to make it look nicer (like $$u=x^3$$ ),but I still can't work out a better substitution.

Could you give me some more advice?

try
$$u=x^3,dv=x^2\cos(x^3)dx$$

have you learned partial substitution ?

mjsd said:
hint: try look for a substitution to make the integrand look nicer, then you can try by parts.

here I suggested a two-step process,
substitiion: u = x^3 seems ok
then by parts in new variable

then put answer back in x.

## What is integration and why is it important?

Integration is a mathematical process that involves finding the area under a curve. It is important because it allows us to solve a variety of real-world problems, such as finding the distance traveled by an object or the amount of water in a tank.

## What is the formula for integration?

The formula for integration is ∫ f(x) dx = F(x) + C, where f(x) is the function to be integrated, F(x) is the antiderivative of f(x), and C is the constant of integration.

## How do I solve the integration of x^5cos(x^3) dx?

To solve this integration, we can use the substitution method. Let u = x^3, then du/dx = 3x^2 and dx = du/3x^2. Substituting this into the original equation, we get ∫ x^5cos(x^3) dx = ∫ u^2cos(u) du. This can be solved using integration by parts or by using the integral table.

## What are the steps for solving an integration problem?

The steps for solving an integration problem are:

1. Identify the function to be integrated.
2. Simplify the function if possible.
3. If the function contains a variable, use a substitution to simplify the problem.
4. Apply integration rules, such as the power rule, product rule, or quotient rule, to find the antiderivative.
5. Add the constant of integration.

## Can integration be solved using a calculator?

Yes, integration can be solved using a graphing calculator or a computer program that has a built-in integration function. However, it is important to understand the steps and concepts behind integration in order to use a calculator effectively and to check the accuracy of the results.

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