First make a substitution and then use integration by parts to evaluate.
The integral of (x^9)(cos(x^5))dx
What do you substitute first? I do not understand what they are asking for. Please help. thanks!
To solve the integral, you need to make a substitution first. Let me guide you through the process step-by-step.
Step 1: Choose a suitable substitution.
In this case, let's substitute u = x^5. This substitution will help simplify the integral and make it easier to solve using integration by parts.
Step 2: Find the derivative of u with respect to x.
Differentiating both sides of the substitution equation u = x^5 with respect to x, we get:
du/dx = 5x^4
Step 3: Solve for dx in terms of du.
Rearranging the equation obtained in step 2, we can solve for dx:
dx = du / (5x^4)
Step 4: Substitute u and dx into the original integral.
Now, substitute u = x^5 and dx = du / (5x^4) into the original integral:
∫ (x^9)(cos(x^5)) dx
= ∫ ((x^5)(x^4))(cos(u)) dx
= ∫ ((1/5)(u)(1/x))(cos(u)) du
Step 5: Simplify the integral.
Using the properties of integration, we can simplify the integral:
(1/5) * ∫ (u/x) * cos(u) du
Step 6: Apply integration by parts.
Now, we can use integration by parts to solve the simplified integral:
∫ u * v' du = u * v - ∫ v * u' du
Choosing u = (1/x) and dv = cos(u) du, we find that:
u' = -1/x^2 and v = sin(u)
Applying the formula for integration by parts, we have:
(1/5) * ∫ (u/x) * cos(u) du
= (1/5) * [(1/x) * sin(u) - ∫ (-1/x^2) * sin(u) du]
= (1/5) * [(1/x) * sin(u) + ∫ (1/x^2) * sin(u) du]
Step 7: Simplify and solve the final integral.
In the last step, we need to evaluate the integral:
∫ (1/x^2) * sin(u) du
This integral can be solved using the reverse of the substitution made earlier:
w = x^5
dw = 5x^4 dx
dx = dw / (5x^4)
Substituting u = x^5 and dx = dw / (5x^4), we get:
∫ (1/x^2) * sin(u) du
= ∫ (1/x^2) * sin(w) * (dw / (5x^4))
= (1/5) * ∫ (1/w^2) * sin(w) dw
This integral can be evaluated using standard integration techniques.
Finally, once you have the result, remember to substitute back u = x^5 into the final expression.
I hope this explanation helps you understand how to solve the integral using substitution and integration by parts. If you have any further questions, feel free to ask!
To evaluate the integral ∫(x^9)(cos(x^5)) dx, we can use a substitution method.
Let's set u = x^5. Now we need to find the derivative of this substitution, du.
Differentiating both sides with respect to x, we get du/dx = d/dx(x^5).
Applying the power rule of differentiation, the derivative of x^5 is 5x^4.
Now, we can solve for dx by rearranging the equation du/dx = 5x^4:
dx = du / 5x^4.
Substituting both the substitution u = x^5 and dx = du / 5x^4 into the original integral, we have:
∫(x^9)(cos(x^5)) dx = ∫(u^(9/5))(cos(u)) (du / 5x^4).
Next, we simplify the integral, taking into account that x^4 = (u^(1/5))^4 = u^(4/5):
∫(u^(9/5))(cos(u)) (du / 5x^4) = (1/5) ∫(u^(9/5))(cos(u)) / u^(4/5) du.
Simplifying further:
(1/5) ∫(u^(9/5 - 4/5))(cos(u)) du = (1/5) ∫(u^1)(cos(u)) du.
Now, we can proceed with integrating u^1(cos(u)):
(1/5) ∫(u^1)(cos(u)) du = (1/5)(∫u du) (by integration by parts).
Integrating u with respect to u gives u^2/2:
(1/5)(∫u du) = (1/5)(u^2/2) + C.
Finally, substituting back the value of u = x^5, our final answer is:
∫(x^9)(cos(x^5)) dx = (1/10)(x^10) + C.
Therefore, the result of the integral is (1/10)(x^10) + C.