Question 3 a) ∫1/x dx b) ∫(3/x + 7x² + 7eˣ – Cos x)dx c) ∫(2x−3)² dx

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Okay, Felix, let's integrate these expressions. a) To evaluate ∫ 1/x dx: Step 1: Recall the standard integral for 1/x. The integral of 1/x with respect to x is the natural logarithm of the absolute value of x. ∫ 1/x dx = ln|x| + C The integral is ln|x| + C. b) To evaluate ∫ (3/x + 7x² + 7e^x − cos x) dx: Step 1: Integrate each term separately using the sum/difference rule for integrals. ∫ (3/x + 7x² + 7e^x − cos x) dx = ∫ 3/x dx + ∫ 7x² dx + ∫ 7e^x dx − ∫ cos x dx; Step 2: Apply the power rule for integration (∫ xⁿ dx = (xⁿ⁺¹)/(n+1)), the integral of e^x, and the integral of cos x. ∫ 3/x dx = 3 ln|x|; ∫ 7x² dx = 7 (x²⁺¹)/(2+1) = 7 (x³)/(3); ∫ 7e^x dx = 7e^x; ∫ cos x dx = sin x Step 3: Combine the results and add the constant of integration, C. ∫ (3/x + 7x² + 7e^x − cos x) dx = 3 ln|x| + 7/3x³ + 7e^x − sin x + C The integral is 3 ln|x| + 7/3x³ + 7e^x − sin x + C. c) To evaluate ∫ (2x−3)² dx: Step 1: Expand the term (2x−3)². (2x−3)² = (2x)² − 2(2x)(3) + (−3)² = 4x² − 12x + 9 Step 2: Integrate the expanded polynomial term by term. ∫ (4x² − 12x + 9) dx = ∫ 4x² dx − ∫ 12x dx + ∫ 9 dx Step 3: Apply the power rule for integration to each term. ∫ 4x² dx = 4 (x²⁺¹)/(2+1) = 4 (x³)/(3); ∫ −12x dx = −12 (x¹⁺¹)/(1+1) = −12 (x²)/(2) = −6x²; ∫ 9 dx = 9x Step 4: Combine the results and add the constant of integration, C. ∫ (2x−3)² dx = 4/3x³ − 6x² + 9x + C The integral is 4/3x³ − 6x² + 9x + C.

Question 3 a) ∫1/x dx b) ∫(3/x + 7x² + 7eˣ – Cos x)dx c) ∫(2x-3)² dx

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Step-by-step solution

Part a· 1 step

Recall the standard integral for $\frac{1}{x}$ .

Part b· 3 steps
$3 \ln|x| + \frac{7}{3}x^3 + 7e^x - \sin x + C$

Integrate each term separately using the sum/difference rule for integrals.

Apply the power rule for integration ( $\int x^n dx = \frac{x^{n+1}}{n+1}$ ), the integral of $e^x$ , and the integral of $\cos x$ .

Combine the results and add the constant of integration, $C$ .

Part c· 4 steps
$\frac{4}{3}x^3 - 6x^2 + 9x + C$

Expand the term $(2x-3)^2$ .

Integrate the expanded polynomial term by term.

Apply the power rule for integration to each term.

Combine the results and add the constant of integration, $C$ .

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Question 3 a) ∫1/x dx b) ∫(3/x + 7x² + 7eˣ – Cos x)dx c) ∫(2x-3)² dx

Answers

Step-by-step solution

Part a· 1 step

Recall the standard integral for 1x\frac{1}{x}x1​.

Part b· 3 steps3ln⁡∣x∣+73x3+7ex−sin⁡x+C3 \ln|x| + \frac{7}{3}x^3 + 7e^x - \sin x + C3ln∣x∣+37​x3+7ex−sinx+C

Integrate each term separately using the sum/difference rule for integrals.

Apply the power rule for integration (∫xndx=xn+1n+1\int x^n dx = \frac{x^{n+1}}{n+1}∫xndx=n+1xn+1​), the integral of exe^xex, and the integral of cos⁡x\cos xcosx.

Combine the results and add the constant of integration, CCC.

Part c· 4 steps43x3−6x2+9x+C\frac{4}{3}x^3 - 6x^2 + 9x + C34​x3−6x2+9x+C

Expand the term (2x−3)2(2x-3)^2(2x−3)2.

Integrate the expanded polynomial term by term.

Apply the power rule for integration to each term.

Combine the results and add the constant of integration, CCC.

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Recall the standard integral for $\frac{1}{x}$ .

Part b· 3 steps
$3 \ln|x| + \frac{7}{3}x^3 + 7e^x - \sin x + C$

Apply the power rule for integration ( $\int x^n dx = \frac{x^{n+1}}{n+1}$ ), the integral of $e^x$ , and the integral of $\cos x$ .

Combine the results and add the constant of integration, $C$ .

Part c· 4 steps
$\frac{4}{3}x^3 - 6x^2 + 9x + C$

Expand the term $(2x-3)^2$ .

Combine the results and add the constant of integration, $C$ .