Integral Evaluation and Substitution Methods

Integral Evaluation Overview

This section presents various integrals to be evaluated using the substitution method, also known as u-substitution. Each integral is denoted with a specific letter for easy reference.

Integral (a)

Evaluate the integral:
$\int \frac{8x}{(x^2 + 5)^4} dx$

Steps for Evaluation

• Substitution: Let ( u = x^2 + 5 ).

• Derivative: Then, ( du = 2x \, dx $or$ dx = \frac{du}{2x} $</p></li><li><p><strong>Change of Variables:</strong> Substitute ( x^2 = u - 5 ), thus the integral transforms into ( \int \frac{8x}{u^4} \frac{du}{2x} = \int \frac{4}{u^4} du ).</p></li><li><p><strong>Integral Result:</strong> The integral can now be evaluated as ( -\frac{4}{3u^3} + C ), where ( C ) is the constant of integration.</p></li><li><p><strong>Back Substitute:</strong> Resubstituting ( u ) gives the final answer.</p></li></ul><h4>Integral (b)</h4><p>Evaluate the integral: <br>$\int (3\cos(3x) + 7) e^{\sin(3x)} \, dx$</p><h5>Steps for Evaluation</h5><ul><li><p><strong>Substitution:</strong> Let ( u = \sin(3x) ), then ( du = 3\cos(3x) \, dx$ or $dx = \frac{du}{3\cos(3x)}).</p></li><li><p><strong>Change of the Integral:</strong> The integral transforms to ( \int (1 + \frac{7e^u}{3\cos(3x)}) du$ alongside the term for ( 3\cos(3x) ).

Integral (c)

Evaluate the integral:
$\int \frac{\sec^2(4x)}{1 + \tan^2(4x)} dx$

Steps for Evaluation

• Identity: Notice that ( 1 + \tan^2(4x) = \sec^2(4x)). Hence, the integral simplifies to ( \int 1 \, dx = x + C ).

Integral (d)

Evaluate the integral:
$\int -4x \, dx$

Steps for Evaluation

• Direct Integration: Integrating directly gives, ( -2x^2 + C ).

Integral (e)

Evaluate the integral:
$\int \frac{1}{\cos(60)}(2 - \sin(60))^7 \, dx$

Steps for Evaluation

• Constant Evaluation: Approximate constants should be calculated. ( \cos(60) = \frac{1}{2}, \sin(60) = \frac{\sqrt{3}}{2}), resulting in a constant factor outside the integral.

Integral (f)

Evaluate the integral:
$\int (2x^2 + 9) \, dx$

Steps for Evaluation

• Direct Integration: This gives: ( \frac{2}{3} x^3 + 9x + C ).

Integral (g)

Evaluate the integral:
$\int \sin^3(t) \cos(t) \, dt$

Steps for Evaluation

• Substitution: Let ( u = \sin(t), du = \cos(t) \, dt).

• Transforming Integral: This leads to the integral ( \int u^3 \, du).

• Evaluating: Results in ( \frac{1}{4} u^4 + C = \frac{1}{4} \sin^4(t) + C ).

Integral (h)

Evaluate the integral:
$\int (3x + 1)^5 \, dx$

Steps for Evaluation

• Substitution: Let ( u = 3x + 1 ), ( du = 3 \, dx), therefore ( dx = \frac{du}{3} .

• Transforming the Integral: The new integral is ( \int u^5 \frac{du}{3} = \frac{1}{18} u^6 + C).

• Back Substitute: Resubstitute for ( u ) to get final answer.

Integral (i)

Evaluate the integral:
$\int \frac{3x^2}{x^3 + 2} \, dx$

Steps for Evaluation

• Substitution: Let ( u = x^3 + 2 \, then ( du = 3x^2 \, dx), thus simply ( \int \frac{1}{u} du = \ln|u| + C).

Integral (j)

Evaluate the integral:
$\int \sin(20) \cos(20) \, dx$

Steps for Evaluation

• Constant Evaluation: Recognize coefficient in integrand as a constant and integrate accordingly.

Integral (k)

Evaluate the integral:
$\int \frac{6}{(2x + 5)^4} \, dx$

Steps for Evaluation

• Substitution: Let ( u = 2x + 5, \quad du = 2dx \quad or \quad dx = \frac{du}{2})

• Evaluating Integral: This becomes ( \int 3u^{-4} du = -\frac{3}{3u^3} + C$.</p></li><li><p><strong>Back Substitute:</strong> Replace ( u) with the original variable for the final answer.</p></li></ul><h4>Integral (l)</h4><p>Evaluate the integral: <br>$\int \frac{\cos(0)}{4 - \sin^2(0)} \, dx$</p><h5>Steps for Evaluation</h5><ul><li><p><strong>Constant Evaluation:</strong> Calculate constants to simplify the integral. Discuss implications of undefined segments in the function analysis.</p></li></ul><h4>Integral (m)</h4><p>Evaluate the integral: <br>$\int \frac{2 \sin(2x)}{3 + \cos(2x)} \, dx$</p><h5>Steps for Evaluation</h5><ul><li><p><strong>Substitution:</strong> Let ( u = \cos(2x); \quad du = -2\sin(2x)dx), leading to ( -\int \frac{1}{3 + u} du).</p></li></ul><h4>Integral (n)</h4><p>Evaluate the integral: <br>$\int \frac{(x + 4)^{3/2}}{5x} \, dx$</p><h5>Steps for Evaluation</h5><ul><li><p><strong>Simplifying Expression:</strong> To simplify the integral, break it down as fractions or equivalent composite terms.</p></li></ul><h4>Integral (o)</h4><p>Evaluate the integral:$ \int (x^2 - 6)^6 \, dx $$

  Steps for Evaluation

  • Direct Integration: Approach through power rule application for polynomial integrals. Provide comprehensive results based on degree.

  Conclusion

  Each integral requires a detailed exploration of the u-substitution method, integrating both the algebraic manipulations and crucial evaluations for accuracy. Make sure to back substitute wherever necessary for deriving the final results accurately.