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Simplify the integral: ∫(3sin x / cos x + π/x + 7) dx

March 26, 2026Mathematics

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Answer

-3 \ln|\cos x| + \pi \ln|x| + 7x + C

5 steps

Separate the integral into individual terms.
$\int \left( \frac{3 \sin x}{\cos x} + \frac{\pi}{x} + 7 \right) dx = \int \frac{3 \sin x}{\cos x} \, dx + \int \frac{\pi}{x} \, dx + \int 7 \, dx$
Integrate the first term.
Recall that $\frac{\sin x}{\cos x} = \tan x$ .
$\int \frac{3 \sin x}{\cos x} \, dx = 3 \int \tan x \, dx$
The integral of $\tan x$ is $-\ln|\cos x|$ .
$3(-\ln|\cos x|) = -3 \ln|\cos x|$
Integrate the second term.
$\int \frac{\pi}{x} \, dx = \pi \int \frac{1}{x} \, dx$
The integral of $\frac{1}{x}$ is $\ln|x|$ .
$\pi \ln|x|$
Integrate the third term.
$\int 7 \, dx = 7x$
Combine all integrated terms and add the constant of integration $C$ .
$-3 \ln|\cos x| + \pi \ln|x| + 7x + C$

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