Evaluate the following integrals: 46. integral ( x)/(3 + x) dx 47. integral (2(2x))/(1 + (2x)) dx

Question

ScanSolve AI Tutor · Accepted Answer

Here are the solutions for the integrals:

:
We need to evaluate the integral $\int \frac{\cos x}{3 + \sin x} dx$ .

Use u-substitution.
Let $u = 3 + \sin x$ .
Find the differential $du$ .
Differentiating $u$ with respect to $x$ gives:
$du = \cos x \, dx$
Substitute $u$ and $du$ into the integral.
The integral becomes:
$\int \frac{1}{u} du$
Integrate with respect to $u$ .
The integral of $\frac{1}{u}$ is $\ln|u|$ .
$\int \frac{1}{u} du = \ln|u| + C$
Substitute back $u = 3 + \sin x$ .
$\ln|3 + \sin x| + C$
The final answer is $\boxed{\ln|3 + \sin x| + C}$ .

:
We need to evaluate the integral $\int \frac{\sec^2(2x)}{1 + \tan(2x)} dx$ .
Use u-substitution.
Let $u = 1 + \tan(2x)$ .
Find the differential $du$ .
Differentiating $u$ with respect to $x$ using the chain rule gives:
$du = \sec^2(2x) \cdot \frac{d}{dx}(2x) \, dx$ $du = \sec^2(2x) \cdot 2 \, dx$
Rearrange to solve for $\sec^2(2x) \, dx$ :
$\frac{1}{2} du = \sec^2(2x) \, dx$
Substitute $u$ and $du$ into the integral.
The integral becomes:
$\int \frac{1}{u} \cdot \frac{1}{2} du$ $\frac{1}{2} \int \frac{1}{u} du$
Integrate with respect to $u$ .
$\frac{1}{2} \ln|u| + C$
Substitute back $u = 1 + \tan(2x)$ .
$\frac{1}{2} \ln|1 + \tan(2x)| + C$
The final answer is $\boxed{\frac{1}{2} \ln|1 + \tan(2x)| + C}$ .

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