Integrate the following functions: (i) integral ( ( x))/(x) dx (ii) integral ( (3 x + 4 x)/(4 x − 3 x) ) dx…

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Integrate the following functions: (i) integral ( ( x))/(x) dx (ii) integral ( (3 x + 4 x)/(4 x − 3 x) ) dx (iii) integral em ^(−1x)1 + x2 dx (iv) integral x (4x2 + 7) dx

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Here are the solutions to the integration problems: (i) To evaluate ∫ (sin(log x))/(x) dx: Step 1: Let u = log x. Then, differentiate u with respect to x: du = 1/x dx Step 2: Substitute u and du into the integral. ∫ sin(u) du Step 3: Integrate with respect to u. −cos(u) + C Step 4: Substitute back u = log x. −cos(log x) + C (ii) To evaluate ∫ ((3 sin x + 4 cos x)/(4 sin x − 3 cos x)) dx: Step 1: Let u = 4 sin x − 3 cos x. Then, differentiate u with respect to x: du = (4 cos x − 3(−sin x)) dx; du = (4 cos x + 3 sin x) dx Step 2: Substitute u and du into the integral. ∫ 1/u du Step 3: Integrate with respect to u. log|u| + C Step 4: Substitute back u = 4 sin x − 3 cos x. log|4 sin x − 3 cos x| + C (iii) To evaluate ∫ e^(m tan⁻¹ x)1+x² dx: Step 1: Let u = m tan⁻¹ x. Then, differentiate u with respect to x: du = m · (1)/(1+x²) dx Rearrange to find (1)/(1+x²) dx: (1)/(1+x²) dx = 1/m du Step 2: Substitute u and du into the integral. ∫ e^u · 1/m du 1/m ∫ e^u du Step 3: Integrate with respect to u. 1/m e^u + C Step 4: Substitute back u = m tan⁻¹ x. 1/m e^(m tan⁻¹ x + C) (iv) To evaluate ∫ x sin (4x² + 7) dx: Step 1: Let u = 4x² + 7. Then, differentiate u with respect to x: du = 8x dx Rearrange to find x dx: x dx = 1/8 du Step 2: Substitute u and du into the integral. ∫ sin(u) · 1/8 du 1/8 ∫ sin(u) du Step 3: Integrate with respect to u. 1/8 (−cos(u)) + C −1/8 cos(u) + C Step 4: Substitute back u = 4x² + 7. −1/8 cos(4x² + 7) + C

Integrate the following functions: (i) integral ( ( x))/(x) dx (ii) integral ( (3 x + 4 x)/(4 x - 3 x) ) dx (iii) integral em ^(-1x)1 + x2 dx (iv) integral x (4x2 + 7) dx

Answers

Step-by-step solution

Part i· 4 steps

Let $u = \log x$ .

Substitute $u$ and $du$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = \log x$ .

Part ii· 4 steps
$\log|4 \sin x - 3 \cos x| + C$

Let $u = 4 \sin x - 3 \cos x$ .

Substitute $u$ and $du$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = 4 \sin x - 3 \cos x$ .

Part iii· 4 steps
$\frac{1}{m} e^{m \tan^{-1} x} + C$

Let $u = m \tan^{-1} x$ .

Substitute $u$ and $du$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = m \tan^{-1} x$ .

Part iv· 4 steps
$-\frac{1}{8} \cos(4x^2 + 7) + C$

Let $u = 4x^2 + 7$ .

Substitute $u$ and $du$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = 4x^2 + 7$ .

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More Mathematics Questions

Integrate the following functions: (i) integral ( ( x))/(x) dx (ii) integral ( (3 x + 4 x)/(4 x - 3 x) ) dx (iii) integral em ^(-1x)1 + x2 dx (iv) integral x (4x2 + 7) dx

Answers

Step-by-step solution

Part i· 4 steps

Let u=log⁡xu = \log xu=logx.

Substitute uuu and dududu into the integral.

Integrate with respect to uuu.

Substitute back u=log⁡xu = \log xu=logx.

Part ii· 4 stepslog⁡∣4sin⁡x−3cos⁡x∣+C\log|4 \sin x - 3 \cos x| + Clog∣4sinx−3cosx∣+C

Let u=4sin⁡x−3cos⁡xu = 4 \sin x - 3 \cos xu=4sinx−3cosx.

Substitute uuu and dududu into the integral.

Integrate with respect to uuu.

Substitute back u=4sin⁡x−3cos⁡xu = 4 \sin x - 3 \cos xu=4sinx−3cosx.

Part iii· 4 steps1memtan⁡−1x+C\frac{1}{m} e^{m \tan^{-1} x} + Cm1​emtan−1x+C

Let u=mtan⁡−1xu = m \tan^{-1} xu=mtan−1x.

Substitute uuu and dududu into the integral.

Integrate with respect to uuu.

Substitute back u=mtan⁡−1xu = m \tan^{-1} xu=mtan−1x.

Part iv· 4 steps−18cos⁡(4x2+7)+C-\frac{1}{8} \cos(4x^2 + 7) + C−81​cos(4x2+7)+C

Let u=4x2+7u = 4x^2 + 7u=4x2+7.

Substitute uuu and dududu into the integral.

Integrate with respect to uuu.

Substitute back u=4x2+7u = 4x^2 + 7u=4x2+7.

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More Mathematics Questions

Let $u = \log x$ .

Substitute $u$ and $du$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = \log x$ .

Part ii· 4 steps
$\log|4 \sin x - 3 \cos x| + C$

Let $u = 4 \sin x - 3 \cos x$ .

Substitute $u$ and $du$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = 4 \sin x - 3 \cos x$ .

Part iii· 4 steps
$\frac{1}{m} e^{m \tan^{-1} x} + C$

Let $u = m \tan^{-1} x$ .

Substitute $u$ and $du$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = m \tan^{-1} x$ .

Part iv· 4 steps
$-\frac{1}{8} \cos(4x^2 + 7) + C$

Let $u = 4x^2 + 7$ .

Substitute $u$ and $du$ into the integral.

Integrate with respect to $u$ .

Substitute back $u = 4x^2 + 7$ .