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Math  /  Calculus

QuestionSolve the following: show your complete solutions 1) ∫(42x+2x2+3)dx\int\left(42 x+2 x^{2}+3\right) d x 2.) ∫2512x43dx\int^{2} \sqrt[3]{512 x^{4}} d x 3.) ∫5x3−43xdx\int \frac{5 x^{3}-4}{3 x} d x

Studdy Solution

STEP 1

1. The integrals involve basic polynomial and rational functions.
2. The result of each integral should be in the simplest form possible.
3. Basic rules of integration and algebraic simplifications will be used.

STEP 2

1. Integrate the polynomial function in part 1.
2. Integrate the function involving a cube root in part 2.
3. Integrate the rational function in part 3.

STEP 3

Integrate the polynomial function 42x+2x2+342x + 2x^2 + 3 term by term. ∫(42x+2x2+3) dx \int(42x + 2x^2 + 3) \, dx

STEP 4

Integrate 42x42x. ∫42x dx=42∫x dx=42(x22)=21x2 \int 42x \, dx = 42 \int x \, dx = 42 \left( \frac{x^2}{2} \right) = 21x^2

STEP 5

Integrate 2x22x^2. ∫2x2 dx=2∫x2 dx=2(x33)=2x33 \int 2x^2 \, dx = 2 \int x^2 \, dx = 2 \left( \frac{x^3}{3} \right) = \frac{2x^3}{3}

STEP 6

Integrate 33. ∫3 dx=3x \int 3 \, dx = 3x

STEP 7

Combine all the integrated terms and add the constant of integration CC. ∫(42x+2x2+3) dx=21x2+2x33+3x+C \int (42x + 2x^2 + 3) \, dx = 21x^2 + \frac{2x^3}{3} + 3x + C

STEP 8

Rewrite the integrand 512x43\sqrt[3]{512 x^4} in a simpler form. 512x43=5121/3⋅x4/3=8⋅x4/3 \sqrt[3]{512 x^4} = 512^{1/3} \cdot x^{4/3} = 8 \cdot x^{4/3}

STEP 9

Integrate 8x4/38x^{4/3}. ∫8x4/3 dx=8∫x4/3 dx=8(37x7/3)=247x7/3 \int 8x^{4/3} \, dx = 8 \int x^{4/3} \, dx = 8 \left( \frac{3}{7} x^{7/3} \right) = \frac{24}{7} x^{7/3}

STEP 10

Evaluate the definite integral from 0 to 2. ∫028x4/3 dx=[247x7/3]02 \int_0^2 8x^{4/3} \, dx = \left[ \frac{24}{7} x^{7/3} \right]_0^2

STEP 11

Substitute the upper limit 22 and the lower limit 00 into the evaluated integral. [247x7/3]02=247(27/3)−247(07/3) \left[ \frac{24}{7} x^{7/3} \right]_0^2 = \frac{24}{7} \left( 2^{7/3} \right) - \frac{24}{7} \left( 0^{7/3} \right) =247⋅27/3=247⋅8=1927 = \frac{24}{7} \cdot 2^{7/3} = \frac{24}{7} \cdot 8 = \frac{192}{7}

STEP 12

Rewrite the integrand 5x3−43x\frac{5x^3 - 4}{3x} in a simpler form by separating the terms. 5x3−43x=5x33x−43x=5x23−43x \frac{5x^3 - 4}{3x} = \frac{5x^3}{3x} - \frac{4}{3x} = \frac{5x^2}{3} - \frac{4}{3x}

STEP 13

Integrate 5x23\frac{5x^2}{3}. ∫5x23 dx=53∫x2 dx=53(x33)=5x39 \int \frac{5x^2}{3} \, dx = \frac{5}{3} \int x^2 \, dx = \frac{5}{3} \left( \frac{x^3}{3} \right) = \frac{5x^3}{9}

STEP 14

Integrate −43x-\frac{4}{3x}. ∫−43x dx=−43∫1x dx=−43ln⁡∣x∣ \int -\frac{4}{3x} \, dx = -\frac{4}{3} \int \frac{1}{x} \, dx = -\frac{4}{3} \ln|x|

STEP 15

Combine all the integrated terms and add the constant of integration CC. ∫(5x23−43x) dx=5x39−43ln⁡∣x∣+C \int \left( \frac{5x^2}{3} - \frac{4}{3x} \right) \, dx = \frac{5x^3}{9} - \frac{4}{3} \ln|x| + C
Solution:
1. ∫(42x+2x2+3)dx=21x2+2x33+3x+C\int(42 x+2 x^{2}+3) d x = 21x^2 + \frac{2x^3}{3} + 3x + C
2. ∫2512x43dx=1927\int^{2} \sqrt[3]{512 x^{4}} d x = \frac{192}{7}
3. ∫5x3−43xdx=5x39−43ln⁡∣x∣+C\int \frac{5 x^{3}-4}{3 x} d x = \frac{5x^3}{9} - \frac{4}{3} \ln|x| + C

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