Math / Calculus

Question(b) Use your work in (a) to help you evaluate the following indefinite integrals. Use differentiation to check your work. (Don't forget the " $+C^{\text {" }}$ .) (i) $\int x e^{x}+e^{x} d x=\left(\mathrm{xe}^{\wedge} \mathrm{x}\right)+\mathrm{C}$ (ii) $\int e^{x}(\sin (x)+\cos (x)) d x=\left(\mathrm{e}^{\wedge} \mathrm{x}\right) \sin (\mathrm{x})+\mathrm{C}$ (iii) $\int 2 x \cos (x)-x^{2} \sin (x) d x=\left(x^{\wedge} 2\right) \sin (x)+2 x \cos (x)-2 \sin (x$ . (iv) $\int x \cos (x)+\sin (x) d x=x \sin (x)+C$ (v) $\int 1+\ln (x) d x=x \ln (\mathrm{x})+\mathrm{C}$ $\qquad$ (c) Observe that the examples in (b) work nicely because of the derivatives you were asked to calculate in (a). Each integrand in (b) is precisely the result of differentiating one of the products of basic functions found in (a). To see what happens when an integrand is still a product but not necessarily the result of differentiating an elementary product, we consider how to evaluate

Studdy Solution

STEP 1

What is this asking? We're checking some integrals by differentiating them, and then we're going to think about why these examples work so well! Watch out! Don't forget the "+C" when integrating, and double-check those minus signs when differentiating!

STEP 2

1. Check Integral (i)
2. Check Integral (ii)
3. Check Integral (iii)
4. Check Integral (iv)
5. Check Integral (v)

STEP 3

We're given $\int x e^{x}+e^{x} d x = xe^x + C$ .
Let's differentiate the right side and see if we get the integrand!

STEP 4

Applying the product rule to $xe^x$ , we get: $\frac{d}{dx} (xe^x) = x \cdot \frac{d}{dx}(e^x) + e^x \cdot \frac{d}{dx}(x) = x \cdot e^x + e^x \cdot 1 = xe^x + e^x.$ The derivative of the constant $C$ is zero, so we have: $\frac{d}{dx}(xe^x + C) = xe^x + e^x.$

STEP 5

This matches the integrand, so the integral is correct!

STEP 6

We have $\int e^{x}(\sin(x)+\cos(x)) dx = e^x \sin(x) + C$ .
Let's differentiate $e^x \sin(x)$ !

STEP 7

Using the product rule: $\frac{d}{dx} (e^x \sin(x)) = e^x \cdot \frac{d}{dx}(\sin(x)) + \sin(x) \cdot \frac{d}{dx}(e^x) = e^x \cos(x) + \sin(x) e^x = e^x (\cos(x) + \sin(x)).$ Adding the derivative of $C$ (which is zero), we get $e^x (\cos(x) + \sin(x))$ .

STEP 8

This matches the integrand, awesome!

STEP 9

We're given $\int 2x \cos(x) - x^2 \sin(x) dx = x^2 \cos(x) + C$ .
Let's differentiate $x^2 \cos(x)$ .

STEP 10

Using the product rule: $\frac{d}{dx}(x^2 \cos(x)) = x^2 \cdot \frac{d}{dx}(\cos(x)) + \cos(x) \cdot \frac{d}{dx}(x^2) = x^2(-\sin(x)) + \cos(x)(2x) = 2x\cos(x) - x^2\sin(x).$ The derivative of C is zero, so the complete derivative is $2x\cos(x) - x^2\sin(x)$ .

STEP 11

It matches!

STEP 12

We have $\int x \cos(x) + \sin(x) dx = x \sin(x) + C$ .
Let's differentiate $x \sin(x)$ .

STEP 13

Using the product rule: $\frac{d}{dx}(x\sin(x)) = x \cdot \frac{d}{dx}(\sin(x)) + \sin(x) \cdot \frac{d}{dx}(x) = x\cos(x) + \sin(x).$ The derivative of $C$ is zero, giving us $x\cos(x) + \sin(x)$ .

STEP 14

Another match!

STEP 15

We're given $\int 1 + \ln(x) dx = x\ln(x) + C$ .
Let's differentiate $x\ln(x)$ .

STEP 16

Product rule time! $\frac{d}{dx}(x\ln(x)) = x \cdot \frac{d}{dx}(\ln(x)) + \ln(x) \cdot \frac{d}{dx}(x) = x \cdot \frac{1}{x} + \ln(x) \cdot 1 = 1 + \ln(x).$ The derivative of $C$ is zero, so we get $1 + \ln(x)$ .

STEP 17

It matches!

STEP 18

All the given integrals are correct when we differentiate the results and add our constant of integration!

Was this helpful?

Studdy Solution

STEP 1

What is this asking? We're checking some integrals by differentiating them, and then we're going to think about why these examples work so well! Watch out! Don't forget the "+C" when integrating, and double-check those minus signs when differentiating!

STEP 2

1. Check Integral (i)2. Check Integral (ii)3. Check Integral (iii)4. Check Integral (iv)5. Check Integral (v)

STEP 3

We're given ∫xex+exdx=xex+C\int x e^{x}+e^{x} d x = xe^x + C∫xex+exdx=xex+C.Let's **differentiate** the right side and see if we get the integrand!

STEP 4

STEP 5

This **matches** the integrand, so the integral is correct!

STEP 6

We have ∫ex(sin⁡(x)+cos⁡(x))dx=exsin⁡(x)+C\int e^{x}(\sin(x)+\cos(x)) dx = e^x \sin(x) + C∫ex(sin(x)+cos(x))dx=exsin(x)+C.Let's **differentiate** exsin⁡(x)e^x \sin(x)exsin(x)!

STEP 7

STEP 8

This **matches** the integrand, awesome!

STEP 9

We're given ∫2xcos⁡(x)−x2sin⁡(x)dx=x2cos⁡(x)+C\int 2x \cos(x) - x^2 \sin(x) dx = x^2 \cos(x) + C∫2xcos(x)−x2sin(x)dx=x2cos(x)+C.Let's **differentiate** x2cos⁡(x)x^2 \cos(x)x2cos(x).

STEP 10

STEP 11

It **matches**!

STEP 12

We have ∫xcos⁡(x)+sin⁡(x)dx=xsin⁡(x)+C\int x \cos(x) + \sin(x) dx = x \sin(x) + C∫xcos(x)+sin(x)dx=xsin(x)+C.Let's **differentiate** xsin⁡(x)x \sin(x)xsin(x).

STEP 13

STEP 14

Another **match**!

STEP 15

We're given ∫1+ln⁡(x)dx=xln⁡(x)+C\int 1 + \ln(x) dx = x\ln(x) + C∫1+ln(x)dx=xln(x)+C.Let's **differentiate** xln⁡(x)x\ln(x)xln(x).

STEP 16

STEP 17

It **matches**!

STEP 18

All the given integrals are correct when we differentiate the results and add our constant of integration!

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

1. Check Integral (i)
2. Check Integral (ii)
3. Check Integral (iii)
4. Check Integral (iv)
5. Check Integral (v)

We're given $\int x e^{x}+e^{x} d x = xe^x + C$ .
Let's differentiate the right side and see if we get the integrand!

This matches the integrand, so the integral is correct!

We have $\int e^{x}(\sin(x)+\cos(x)) dx = e^x \sin(x) + C$ .
Let's differentiate $e^x \sin(x)$ !

This matches the integrand, awesome!

We're given $\int 2x \cos(x) - x^2 \sin(x) dx = x^2 \cos(x) + C$ .
Let's differentiate $x^2 \cos(x)$ .

It matches!

We have $\int x \cos(x) + \sin(x) dx = x \sin(x) + C$ .
Let's differentiate $x \sin(x)$ .

Another match!

We're given $\int 1 + \ln(x) dx = x\ln(x) + C$ .
Let's differentiate $x\ln(x)$ .

It matches!