Evaluate the followings: 1. sqrt(7x−1) dx 2. sin(4x + 7) dx 3. sqrt(sin x) cos x dx 4. x cos x dx 5. e^x sin…

Question

Evaluate the followings: 1. sqrt(7x−1) dx 2. sin(4x + 7) dx 3. sqrt(sin x) cos x dx 4. x cos x dx 5. e^x sin x dx

ScanSolve AI Tutor · Accepted Answer

Here are the evaluations of the integrals:

For $\int \sqrt{7x-1}\,dx$ :

Let $u = 7x-1$ .
Then $du = 7\,dx$ , which means $dx = \frac{1}{7}\,du$ . $\int \sqrt{u} \cdot \frac{1}{7}\,du = \frac{1}{7} \int u^{1/2}\,du$

Integrate $u^{1/2}$ .

$$
\frac{1}{7} \cdot \frac{u^{3/2}}{3/2} + C = \frac{1}{7} \cdot \frac{2}{3} u^{3/2} + C = \frac{2}{21} u^{3/2} + C
$$

Substitute back $u = 7x-1$ .
$$ \boxed{\frac{2}{21} (7x-1)^{3/2} + C} $$
2. For $\int \sin(4\pi x + 7)\,dx$ :
Let $u = 4\pi x + 7$ .
Then $du = 4\pi\,dx$ , which means $dx = \frac{1}{4\pi}\,du$ . $\int \sin(u) \cdot \frac{1}{4\pi}\,du = \frac{1}{4\pi} \int \sin(u)\,du$

Integrate $\sin(u)$ .

$$
\frac{1}{4\pi} (-\cos(u)) + C = -\frac{1}{4\pi} \cos(u) + C
$$

Substitute back $u = 4\pi x + 7$ .
$$ \boxed{-\frac{1}{4\pi} \cos(4\pi x + 7) + C} $$
3. For $\int \sqrt{\sin x} \cos x\,dx$ :
Let $u = \sin x$ .
Then $du = \cos x\,dx$ . $\int \sqrt{u}\,du = \int u^{1/2}\,du$

Integrate $u^{1/2}$ .

$$
\frac{u^{3/2}}{3/2} + C = \frac{2}{3} u^{3/2} + C
$$

Substitute back $u = \sin x$ .
$$ \boxed{\frac{2}{3} (\sin x)^{3/2} + C} $$
4. For $\int x \cos x\,dx$ :
Use integration by parts, $\int v\,dw = vw - \int w\,dv$ .
Let $v = x$ and $dw = \cos x\,dx$ .
Differentiate $v$ and integrate $dw$ .
So, $dv = dx$ and $w = \sin x$ . $\int x \cos x\,dx = x \sin x - \int \sin x\,dx$
Integrate $\sin x$ .
$$ x \sin x - (-\cos x) + C = x \sin x + \cos x + C $$ $$ \boxed{x \sin x + \cos x + C} $$
5. For $\int e^x \sin x\,dx$ :
Use integration by parts.
Let $I = \int e^x \sin x\,dx$ . Let $u = \sin x$ and $dv = e^x\,dx$ .
Differentiate $u$ and integrate $dv$ .
So, $du = \cos x\,dx$ and $v = e^x$ . $I = e^x \sin x - \int e^x \cos x\,dx$
Apply integration by parts again to $\int e^x \cos x\,dx$ .
Let $u = \cos x$ and $dv = e^x\,dx$ .
Differentiate $u$ and integrate $dv$ .
So, $du = -\sin x\,dx$ and $v = e^x$ . $\int e^x \cos x\,dx = e^x \cos x - \int e^x (-\sin x)\,dx = e^x \cos x + \int e^x \sin x\,dx$

Substitute this back into the expression for $I$ .

$$
I = e^x \sin x - (e^x \cos x + \int e^x \sin x\,dx)
$$
$$
I = e^x \sin x - e^x \cos x - I
$$

Solve for $I$ .

$$
2I = e^x \sin x - e^x \cos x
$$
$$
I = \frac{e^x}{2} (\sin x - \cos x) + C
$$
$$
\boxed{\frac{e^x}{2} (\sin x - \cos x) + C}
$$

Evaluate the followings: 1. sqrt(7x-1) dx 2. sin(4x + 7) dx 3. sqrt(sin x) cos x dx 4. x cos x dx 5. e^x sin x dx

Step-by-step solution

Let $u = 7x-1$ .

Integrate $u^{1/2}$ .

Substitute back $u = 7x-1$ .

Let $u = 4\pi x + 7$ .

Integrate $\sin(u)$ .

Substitute back $u = 4\pi x + 7$ .

Let $u = \sin x$ .

Integrate $u^{1/2}$ .

Substitute back $u = \sin x$ .

Use integration by parts, $\int v\,dw = vw - \int w\,dv$ .

Differentiate $v$ and integrate $dw$ .

Integrate $\sin x$ .

Use integration by parts.

Differentiate $u$ and integrate $dv$ .

Apply integration by parts again to $\int e^x \cos x\,dx$ .

Differentiate $u$ and integrate $dv$ .

Substitute this back into the expression for $I$ .

Solve for $I$ .

Still stuck on this one?

More Mathematics Questions

Evaluate the followings: 1. sqrt(7x-1) dx 2. sin(4x + 7) dx 3. sqrt(sin x) cos x dx 4. x cos x dx 5. e^x sin x dx

Step-by-step solution

Let u=7x−1u = 7x-1u=7x−1.

Integrate u1/2u^{1/2}u1/2.

Substitute back u=7x−1u = 7x-1u=7x−1.

Let u=4πx+7u = 4\pi x + 7u=4πx+7.

Integrate sin⁡(u)\sin(u)sin(u).

Substitute back u=4πx+7u = 4\pi x + 7u=4πx+7.

Let u=sin⁡xu = \sin xu=sinx.

Integrate u1/2u^{1/2}u1/2.

Substitute back u=sin⁡xu = \sin xu=sinx.

Use integration by parts, ∫v dw=vw−∫w dv\int v\,dw = vw - \int w\,dv∫vdw=vw−∫wdv.

Differentiate vvv and integrate dwdwdw.

Integrate sin⁡x\sin xsinx.

Use integration by parts.

Differentiate uuu and integrate dvdvdv.

Apply integration by parts again to ∫excos⁡x dx\int e^x \cos x\,dx∫excosxdx.

Differentiate uuu and integrate dvdvdv.

Substitute this back into the expression for III.

Solve for III.

Still stuck on this one?

More Mathematics Questions

Let $u = 7x-1$ .

Integrate $u^{1/2}$ .

Substitute back $u = 7x-1$ .

Let $u = 4\pi x + 7$ .

Integrate $\sin(u)$ .

Substitute back $u = 4\pi x + 7$ .

Let $u = \sin x$ .

Integrate $u^{1/2}$ .

Substitute back $u = \sin x$ .

Use integration by parts, $\int v\,dw = vw - \int w\,dv$ .

Differentiate $v$ and integrate $dw$ .

Integrate $\sin x$ .

Differentiate $u$ and integrate $dv$ .

Apply integration by parts again to $\int e^x \cos x\,dx$ .

Differentiate $u$ and integrate $dv$ .

Substitute this back into the expression for $I$ .

Solve for $I$ .