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

March 26, 2026Mathematics

ScanSolve AI Answer

March 26, 2026

Answer

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

Step 1: Separate the integral into individual terms. $\int \left( \frac{3 \sin x}{\cos x} + \frac{\pi}{x} + 7 \right) dx = \int \frac{3 \sin x}{\cos x} \, dx + \int \frac{\pi}{x} \, dx + \int 7 \, dx$

Step 2: 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 $-\ln|\cos x|$ . $3(-\ln|\cos x|) = -3 \ln|\cos x|$

Step 3: Integrate the second term. $\int \frac{\pi}{x} \, dx = \pi \int \frac{1}{x} \, dx$ The integral of $\frac{1}{x}$ is $\ln|x|$ . $\pi \ln|x|$

Step 4: Integrate the third term. $\int 7 \, dx = 7x$

Step 5: Combine all integrated terms and add the constant of integration $C$ . $-3 \ln|\cos x| + \pi \ln|x| + 7x + C$ The final answer is $\boxed{-3 \ln|\cos x| + \pi \ln|x| + 7x + C}$ .

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Quick Answer

Separate the integral into individual terms. ( (3 x)/( x) + ()/(x) + 7 ) dx = (3 x)/( x) \, dx + ()/(x) \, dx + 7 \, dx Step 2: Integrate the first term.

Asked on March 26, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

ScanSolve AI AnswerAnswered on March 26, 2026

Step 1: Separate the integral into individual terms. ( (3 x)/( x) + ()/(x) + 7 ) dx = (3 x)/( x) \, dx + ()/(x) \, dx + 7 \, dx Step 2: Integrate the first term. Recall that ( x)/( x) = x. (3 x)/( x) \, dx = 3 x \, dx The integral of x is -| x|. 3(-| x|) = -3 | x| Step 3: Integrate the second term. ()/(x) \, dx = (1)/(x) \, dx The integral of (1)/(x) is |x|. |x| Step 4: Integrate the third term. 7 \, dx = 7x Step 5: Combine all integrated terms and add the constant of integration C. -3 | x| + |x| + 7x + C The final answer is -3 | x| + |x| + 7x + C.