A student evaluates the indefinite integral of a polynomial load function. Given coefficient (a) = 5.0000; exponent (n) = 4.0000; x-value (x) = 3.0000; constant of integration (C) = 4.5000, determine the antiderivative value (F).

Start with the thinking

• The governing relation printed in this handbook section is Indefinite integrals — power rule.
• Everything except F is given, so isolate F symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule indefinite integral gives the antiderivative of a x^n plus a constant of integration.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

2. Step 2 — Rearrange symbolically for F:

   $$F = \dfrac{a}{n+1}x^{n+1} + C$$

3. Step 3 — List the givens: coefficient (a) = 5.0000, exponent (n) = 4.0000, x-value (x) = 3.0000, constant of integration (C) = 4.5000.

4. Step 4 — Substitute the given values:

   $$F = \dfrac{5.0000}{4.0000+1}3.0000^{4.0000+1} + 4.5000$$

5. Step 5 — Evaluate:

   $$F = 247.5$$

6. Step 6 — Check: returning F = 247.5 to

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 495.0 — kept a factor of two that cancels in the correct rearrangement.
• 123.8 — dropped that same factor in the other direction.
• 272.3 — rounded an intermediate value before the final step.

Reference: FE Handbook — Indefinite Integrals

An engineer integrates a velocity function to obtain the indefinite integral for displacement. Given coefficient (a) = 4.8000; exponent (n) = 3.0000; x-value (x) = 1.3000; antiderivative value (F) = 240.5, determine the constant of integration (C).

Start with the thinking

• The governing relation printed in this handbook section is Indefinite integrals — power rule.
• Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule indefinite integral gives the antiderivative of a x^n plus a constant of integration.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

2. Step 2 — Rearrange symbolically for C:

   $$C = F - \dfrac{a}{n+1}x^{n+1}$$

3. Step 3 — List the givens: coefficient (a) = 4.8000, exponent (n) = 3.0000, x-value (x) = 1.3000, antiderivative value (F) = 240.5.

4. Step 4 — Substitute the given values:

   $$C = 240.5 - \dfrac{4.8000}{3.0000+1}1.3000^{3.0000+1}$$

5. Step 5 — Evaluate:

   $$C = 237.1$$

6. Step 6 — Check: returning C = 237.1 to

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 474.1 — kept a factor of two that cancels in the correct rearrangement.
• 118.5 — dropped that same factor in the other direction.
• 260.8 — rounded an intermediate value before the final step.

Reference: FE Handbook — Indefinite Integrals

The indefinite integral of an acceleration function is checked against a known constant. Given exponent (n) = 1.0000; x-value (x) = 3.7000; constant of integration (C) = 1.7000; antiderivative value (F) = 276.6, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Indefinite integrals — power rule.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule indefinite integral gives the antiderivative of a x^n plus a constant of integration.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{(F-C)(n+1)}{x^{n+1}}$$

3. Step 3 — List the givens: exponent (n) = 1.0000, x-value (x) = 3.7000, constant of integration (C) = 1.7000, antiderivative value (F) = 276.6.

4. Step 4 — Substitute the given values:

   $$a = \dfrac{(276.6-1.7000)(1.0000+1)}{3.7000^{1.0000+1}}$$

5. Step 5 — Evaluate:

   $$a = 40.1607$$

6. Step 6 — Check: returning a = 40.1607 to

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 80.3214 — kept a factor of two that cancels in the correct rearrangement.
• 20.0804 — dropped that same factor in the other direction.
• 44.1768 — rounded an intermediate value before the final step.

Reference: FE Handbook — Indefinite Integrals

A student evaluates the indefinite integral of a polynomial load function. Given coefficient (a) = 3.1000; exponent (n) = 3.0000; x-value (x) = 3.0000; constant of integration (C) = 4.8000, determine the antiderivative value (F).

Start with the thinking

• The governing relation printed in this handbook section is Indefinite integrals — power rule.
• Everything except F is given, so isolate F symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule indefinite integral gives the antiderivative of a x^n plus a constant of integration.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

2. Step 2 — Rearrange symbolically for F:

   $$F = \dfrac{a}{n+1}x^{n+1} + C$$

3. Step 3 — List the givens: coefficient (a) = 3.1000, exponent (n) = 3.0000, x-value (x) = 3.0000, constant of integration (C) = 4.8000.

4. Step 4 — Substitute the given values:

   $$F = \dfrac{3.1000}{3.0000+1}3.0000^{3.0000+1} + 4.8000$$

5. Step 5 — Evaluate:

   $$F = 67.5750$$

6. Step 6 — Check: returning F = 67.5750 to

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 135.2 — kept a factor of two that cancels in the correct rearrangement.
• 33.7875 — dropped that same factor in the other direction.
• 74.3325 — rounded an intermediate value before the final step.

Reference: FE Handbook — Indefinite Integrals

An engineer integrates a velocity function to obtain the indefinite integral for displacement. Given coefficient (a) = 3.2000; exponent (n) = 4.0000; x-value (x) = 2.7000; antiderivative value (F) = 468.8, determine the constant of integration (C).

Start with the thinking

• The governing relation printed in this handbook section is Indefinite integrals — power rule.
• Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule indefinite integral gives the antiderivative of a x^n plus a constant of integration.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

2. Step 2 — Rearrange symbolically for C:

   $$C = F - \dfrac{a}{n+1}x^{n+1}$$

3. Step 3 — List the givens: coefficient (a) = 3.2000, exponent (n) = 4.0000, x-value (x) = 2.7000, antiderivative value (F) = 468.8.

4. Step 4 — Substitute the given values:

   $$C = 468.8 - \dfrac{3.2000}{4.0000+1}2.7000^{4.0000+1}$$

5. Step 5 — Evaluate:

   $$C = 377.0$$

6. Step 6 — Check: returning C = 377.0 to

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 753.9 — kept a factor of two that cancels in the correct rearrangement.
• 188.5 — dropped that same factor in the other direction.
• 414.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Indefinite Integrals

The indefinite integral of an acceleration function is checked against a known constant. Given exponent (n) = 3.0000; x-value (x) = 3.5000; constant of integration (C) = -3.4000; antiderivative value (F) = 142.3, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Indefinite integrals — power rule.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule indefinite integral gives the antiderivative of a x^n plus a constant of integration.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{(F-C)(n+1)}{x^{n+1}}$$

3. Step 3 — List the givens: exponent (n) = 3.0000, x-value (x) = 3.5000, constant of integration (C) = -3.4000, antiderivative value (F) = 142.3.

4. Step 4 — Substitute the given values:

   $$a = \dfrac{(142.3--3.4000)(3.0000+1)}{3.5000^{3.0000+1}}$$

5. Step 5 — Evaluate:

   $$a = 3.8837$$

6. Step 6 — Check: returning a = 3.8837 to

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 7.7674 — kept a factor of two that cancels in the correct rearrangement.
• 1.9419 — dropped that same factor in the other direction.
• 4.2721 — rounded an intermediate value before the final step.

Reference: FE Handbook — Indefinite Integrals

A student evaluates the indefinite integral of a polynomial load function. Given coefficient (a) = 1.7000; exponent (n) = 4.0000; x-value (x) = 2.0000; constant of integration (C) = 0.8000, determine the antiderivative value (F).

Start with the thinking

• The governing relation printed in this handbook section is Indefinite integrals — power rule.
• Everything except F is given, so isolate F symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule indefinite integral gives the antiderivative of a x^n plus a constant of integration.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

2. Step 2 — Rearrange symbolically for F:

   $$F = \dfrac{a}{n+1}x^{n+1} + C$$

3. Step 3 — List the givens: coefficient (a) = 1.7000, exponent (n) = 4.0000, x-value (x) = 2.0000, constant of integration (C) = 0.8000.

4. Step 4 — Substitute the given values:

   $$F = \dfrac{1.7000}{4.0000+1}2.0000^{4.0000+1} + 0.8000$$

5. Step 5 — Evaluate:

   $$F = 11.6800$$

6. Step 6 — Check: returning F = 11.6800 to

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 23.3600 — kept a factor of two that cancels in the correct rearrangement.
• 5.8400 — dropped that same factor in the other direction.
• 12.8480 — rounded an intermediate value before the final step.

Reference: FE Handbook — Indefinite Integrals

An engineer integrates a velocity function to obtain the indefinite integral for displacement. Given coefficient (a) = 4.8000; exponent (n) = 3.0000; x-value (x) = 1.5000; antiderivative value (F) = 468.8, determine the constant of integration (C).

Start with the thinking

• The governing relation printed in this handbook section is Indefinite integrals — power rule.
• Everything except C is given, so isolate C symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule indefinite integral gives the antiderivative of a x^n plus a constant of integration.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

2. Step 2 — Rearrange symbolically for C:

   $$C = F - \dfrac{a}{n+1}x^{n+1}$$

3. Step 3 — List the givens: coefficient (a) = 4.8000, exponent (n) = 3.0000, x-value (x) = 1.5000, antiderivative value (F) = 468.8.

4. Step 4 — Substitute the given values:

   $$C = 468.8 - \dfrac{4.8000}{3.0000+1}1.5000^{3.0000+1}$$

5. Step 5 — Evaluate:

   $$C = 462.7$$

6. Step 6 — Check: returning C = 462.7 to

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 925.5 — kept a factor of two that cancels in the correct rearrangement.
• 231.4 — dropped that same factor in the other direction.
• 509.0 — rounded an intermediate value before the final step.

Reference: FE Handbook — Indefinite Integrals

The indefinite integral of an acceleration function is checked against a known constant. Given exponent (n) = 2.0000; x-value (x) = 1.1000; constant of integration (C) = -1.9000; antiderivative value (F) = 111.7, determine the coefficient (a).

Start with the thinking

• The governing relation printed in this handbook section is Indefinite integrals — power rule.
• Everything except a is given, so isolate a symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule indefinite integral gives the antiderivative of a x^n plus a constant of integration.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

2. Step 2 — Rearrange symbolically for a:

   $$a = \dfrac{(F-C)(n+1)}{x^{n+1}}$$

3. Step 3 — List the givens: exponent (n) = 2.0000, x-value (x) = 1.1000, constant of integration (C) = -1.9000, antiderivative value (F) = 111.7.

4. Step 4 — Substitute the given values:

   $$a = \dfrac{(111.7--1.9000)(2.0000+1)}{1.1000^{2.0000+1}}$$

5. Step 5 — Evaluate:

   $$a = 256.0$$

6. Step 6 — Check: returning a = 256.0 to

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 512.1 — kept a factor of two that cancels in the correct rearrangement.
• 128.0 — dropped that same factor in the other direction.
• 281.7 — rounded an intermediate value before the final step.

Reference: FE Handbook — Indefinite Integrals

A student evaluates the indefinite integral of a polynomial load function. Given coefficient (a) = 4.4000; exponent (n) = 2.0000; x-value (x) = 1.0000; constant of integration (C) = 4.6000, determine the antiderivative value (F).

Start with the thinking

• The governing relation printed in this handbook section is Indefinite integrals — power rule.
• Everything except F is given, so isolate F symbolically first — never rearrange after the numbers are in.
• Tabulate each given with its unit and confirm the units are consistent with the relation before substituting.
• The power-rule indefinite integral gives the antiderivative of a x^n plus a constant of integration.

Step-by-step solution

1. Step 1 — State the governing relation:

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

2. Step 2 — Rearrange symbolically for F:

   $$F = \dfrac{a}{n+1}x^{n+1} + C$$

3. Step 3 — List the givens: coefficient (a) = 4.4000, exponent (n) = 2.0000, x-value (x) = 1.0000, constant of integration (C) = 4.6000.

4. Step 4 — Substitute the given values:

   $$F = \dfrac{4.4000}{2.0000+1}1.0000^{2.0000+1} + 4.6000$$

5. Step 5 — Evaluate:

   $$F = 6.0667$$

6. Step 6 — Check: returning F = 6.0667 to

   $$\int a x^n \, dx = \dfrac{a}{n+1} x^{n+1} + C$$

   reproduces the given quantities, and both sides carry the same units.

Why the other options are there

• 12.1333 — kept a factor of two that cancels in the correct rearrangement.
• 3.0333 — dropped that same factor in the other direction.
• 6.6733 — rounded an intermediate value before the final step.

Reference: FE Handbook — Indefinite Integrals