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Math  /  Calculus

Question[6] Evaluate the following: ∫(x+8x3+1e5x+1x+2+8)dx\int\left(\sqrt{x}+8 x^{3}+\frac{1}{e^{5 x}}+\frac{1}{x+2}+8\right) d x

Studdy Solution

STEP 1

1. The integral is indefinite, and we are finding the antiderivative of the function.
2. The function is a sum of terms, each of which can be integrated separately.
3. Basic integration rules and techniques, such as power rule and substitution, will be used.

STEP 2

1. Integrate each term separately.
2. Combine the results to find the overall antiderivative.
3. Add the constant of integration.

STEP 3

Integrate the first term x\sqrt{x}, which can be rewritten as x1/2x^{1/2}:
∫x dx=∫x1/2 dx\int \sqrt{x} \, dx = \int x^{1/2} \, dx
Using the power rule for integration, ∫xn dx=xn+1n+1+C\int x^n \, dx = \frac{x^{n+1}}{n+1} + C, we get:
∫x1/2 dx=x1/2+11/2+1=x3/23/2=23x3/2\int x^{1/2} \, dx = \frac{x^{1/2 + 1}}{1/2 + 1} = \frac{x^{3/2}}{3/2} = \frac{2}{3} x^{3/2}

STEP 4

Integrate the second term 8x38x^3:
∫8x3 dx\int 8x^3 \, dx
Using the power rule:
=8⋅x3+13+1=8⋅x44=2x4= 8 \cdot \frac{x^{3+1}}{3+1} = 8 \cdot \frac{x^4}{4} = 2x^4

STEP 5

Integrate the third term 1e5x\frac{1}{e^{5x}}, which can be rewritten as e−5xe^{-5x}:
∫1e5x dx=∫e−5x dx\int \frac{1}{e^{5x}} \, dx = \int e^{-5x} \, dx
Using the rule ∫eax dx=1aeax+C\int e^{ax} \, dx = \frac{1}{a} e^{ax} + C, we get:
=1−5e−5x=−15e−5x= \frac{1}{-5} e^{-5x} = -\frac{1}{5} e^{-5x}

STEP 6

Integrate the fourth term 1x+2\frac{1}{x+2}:
∫1x+2 dx\int \frac{1}{x+2} \, dx
Using the rule ∫1u du=ln⁡∣u∣+C\int \frac{1}{u} \, du = \ln |u| + C, we get:
=ln⁡∣x+2∣= \ln |x+2|

STEP 7

Integrate the fifth term 88:
∫8 dx\int 8 \, dx
This is a constant, so:
=8x= 8x

STEP 8

Combine all the integrated terms:
∫(x+8x3+1e5x+1x+2+8) dx=23x3/2+2x4−15e−5x+ln⁡∣x+2∣+8x+C\int \left(\sqrt{x} + 8x^3 + \frac{1}{e^{5x}} + \frac{1}{x+2} + 8\right) \, dx = \frac{2}{3} x^{3/2} + 2x^4 - \frac{1}{5} e^{-5x} + \ln |x+2| + 8x + C
The antiderivative of the given function is:
23x3/2+2x4−15e−5x+ln⁡∣x+2∣+8x+C\boxed{\frac{2}{3} x^{3/2} + 2x^4 - \frac{1}{5} e^{-5x} + \ln |x+2| + 8x + C}

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