Step-by-Step Guidance for Calculus II: Hyperbolic Functions, Derivatives, and Integrals

Q1. Find the derivative of $y = \sinh^2(6x)$

Background

Topic: Differentiation of Hyperbolic Functions & Chain Rule

This question tests your ability to differentiate a function involving the square of the hyperbolic sine function, with a composite argument. You will need to use the chain rule and the derivative of $\sinh(x)$.

Key Terms and Formulas

• $\sinh(x)$: Hyperbolic sine function

• Derivative: $\frac{d}{dx} \sinh(x) = \cosh(x)$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

• Power Rule: $\frac{d}{dx} [u(x)]^n = n[u(x)]^{n-1} u'(x)$

Step-by-Step Guidance

1. Recognize that $y = [\sinh(6x)]^2$ is a composite function: an outer square and an inner hyperbolic sine.

2. Apply the power rule: $\frac{d}{dx} [u(x)]^2 = 2u(x)u'(x)$, where $u(x) = \sinh(6x)$.

3. Find $u'(x)$: Differentiate $\sinh(6x)$ with respect to $x$ using the chain rule. Recall $\frac{d}{dx} \sinh(ax) = a\cosh(ax)$.

4. Combine the results to express $y'$ in terms of $\sinh(6x)$ and $\cosh(6x)$, but do not simplify to a final numeric expression yet.

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Q2. Find the derivative of $y = \ln(\sinh(7x))$

Background

Topic: Differentiation of Logarithmic and Hyperbolic Functions

This question tests your ability to differentiate a natural logarithm of a hyperbolic sine function. You will need to use the chain rule and the derivative of $\ln(x)$ and $\sinh(x)$.

Key Terms and Formulas

• Derivative: $\frac{d}{dx} \ln(u(x)) = \frac{u'(x)}{u(x)}$

• $\frac{d}{dx} \sinh(ax) = a\cosh(ax)$

Step-by-Step Guidance

1. Let $u(x) = \sinh(7x)$. The derivative of $\ln(u(x))$ is $\frac{u'(x)}{u(x)}$.

2. Find $u'(x)$: Differentiate $\sinh(7x)$ with respect to $x$ using the chain rule.

3. Substitute $u(x)$ and $u'(x)$ into the formula $\frac{u'(x)}{u(x)}$.

4. Simplify the expression as much as possible, but do not compute the final answer.

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Q3. Find the derivative of $y = \cosh^{-1}(2x + 9)$

Background

Topic: Differentiation of Inverse Hyperbolic Functions

This question tests your ability to differentiate the inverse hyperbolic cosine function with a linear argument. You will need to use the chain rule and the derivative of $\cosh^{-1}(x)$.

Key Terms and Formulas

• Derivative: $\frac{d}{dx} \cosh^{-1}(x) = \frac{1}{\sqrt{x^2 - 1}}$

• Chain Rule: $\frac{d}{dx} f(g(x)) = f'(g(x)) \cdot g'(x)$

Step-by-Step Guidance

1. Let $u(x) = 2x + 9$. The derivative of $\cosh^{-1}(u(x))$ is $\frac{1}{\sqrt{u(x)^2 - 1}} \cdot u'(x)$.

2. Compute $u'(x)$, the derivative of $2x + 9$ with respect to $x$.

3. Substitute $u(x)$ and $u'(x)$ into the formula for the derivative.

4. Simplify the expression, but do not compute the final answer.

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Q4. Find the derivative of $y = (1 - 3t) \coth^{-1}(3t)$ with respect to $t$

Background

Topic: Product Rule and Differentiation of Inverse Hyperbolic Functions

This question tests your ability to use the product rule and differentiate the inverse hyperbolic cotangent function.

Key Terms and Formulas

• Product Rule: $\frac{d}{dt}[u(t)v(t)] = u'(t)v(t) + u(t)v'(t)$

• Derivative: $\frac{d}{dt} \coth^{-1}(x) = \frac{-1}{1 - x^2}$

• Chain Rule: $\frac{d}{dt} \coth^{-1}(3t) = \frac{-1}{1 - (3t)^2} \cdot 3$

Step-by-Step Guidance

1. Let $u(t) = 1 - 3t$ and $v(t) = \coth^{-1}(3t)$. Apply the product rule.

2. Compute $u'(t)$, the derivative of $1 - 3t$ with respect to $t$.

3. Compute $v'(t)$, the derivative of $\coth^{-1}(3t)$ with respect to $t$ using the chain rule.

4. Combine the results using the product rule, but do not simplify to a final answer.

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Q5. Evaluate $\int \sinh(4x - \ln 4)\,dx$

Background

Topic: Integration of Hyperbolic Functions

This question tests your ability to integrate a hyperbolic sine function with a linear argument. You will need to use substitution and the integral of $\sinh(ax + b)$.

Key Terms and Formulas

• Integral: $\int \sinh(ax + b)\,dx = \frac{1}{a}\cosh(ax + b) + C$

Step-by-Step Guidance

1. Identify $a$ and $b$ in $\sinh(ax + b)$: here, $a = 4$, $b = -\ln 4$.

2. Recall the formula for integrating $\sinh(ax + b)$.

3. Apply the formula, substituting $a$ and $b$ into the result, but do not simplify to a final answer.

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Q6. Evaluate $\int \sech^2(2x - 7)\,dx$

Background

Topic: Integration of Hyperbolic Functions

This question tests your ability to integrate the square of the hyperbolic secant function with a linear argument. You will need to use substitution and the integral of $\sech^2(ax + b)$.

Key Terms and Formulas

• Integral: $\int \sech^2(ax + b)\,dx = \frac{1}{a}\tanh(ax + b) + C$

Step-by-Step Guidance

1. Identify $a$ and $b$ in $\sech^2(ax + b)$: here, $a = 2$, $b = -7$.

2. Recall the formula for integrating $\sech^2(ax + b)$.

3. Apply the formula, substituting $a$ and $b$ into the result, but do not simplify to a final answer.

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Q7. Evaluate $\int_{\ln 2}^{\ln 7} \tanh x\,dx$

Background

Topic: Definite Integrals of Hyperbolic Functions

This question tests your ability to compute a definite integral involving the hyperbolic tangent function. You will need to recall the antiderivative of $\tanh x$ and evaluate it at the bounds.

Key Terms and Formulas

• Integral: $\int \tanh x\,dx = \ln|\cosh x| + C$

Step-by-Step Guidance

1. Recall the antiderivative of $\tanh x$.

2. Set up the definite integral using the Fundamental Theorem of Calculus: $F(b) - F(a)$, where $F(x)$ is the antiderivative.

3. Substitute the upper and lower bounds ($\ln 7$ and $\ln 2$) into the antiderivative, but do not compute the final value.

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Q8. Evaluate $\int_0^{\ln 2} e^{-t} \sinh t\,dt$

Background

Topic: Definite Integrals and Integration by Parts

This question tests your ability to integrate a product of exponential and hyperbolic sine functions. You may use integration by parts or rewrite $\sinh t$ in terms of exponentials.

Key Terms and Formulas

• $\sinh t = \frac{e^{t} - e^{-t}}{2}$

• Integration by parts: $\int u\,dv = uv - \int v\,du$

Step-by-Step Guidance

1. Rewrite $\sinh t$ in terms of exponentials to simplify the integrand.

2. Multiply $e^{-t}$ by the rewritten form of $\sinh t$ to combine the exponentials.

3. Simplify the integrand and split into two simpler integrals if possible.

4. Set up the definite integrals and prepare to evaluate at the bounds, but do not compute the final value.

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Q9. Evaluate $\int \frac{dx}{25 + x^2}$ in terms of an inverse hyperbolic function

Background

Topic: Integration and Inverse Hyperbolic Functions

This question tests your ability to recognize the integral as a standard form that results in an inverse hyperbolic sine function.

Key Terms and Formulas

• Integral: $\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \arctan\left(\frac{x}{a}\right) + C$

• Alternatively, $\int \frac{dx}{a^2 + x^2} = \frac{1}{a} \sinh^{-1}\left(\frac{x}{a}\right) + C$

Step-by-Step Guidance

1. Identify $a$ in the denominator: here, $a = 5$.

2. Recall the formula for the integral in terms of $\sinh^{-1}$.

3. Apply the formula, substituting $a$ and $x$ into the result, but do not simplify to a final answer.

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Q10. Evaluate $\int_4^8 \frac{dx}{x^2 - 9}$ in terms of natural logarithms

Background

Topic: Partial Fractions and Definite Integrals

This question tests your ability to integrate a rational function using partial fractions and express the result in terms of natural logarithms.

Key Terms and Formulas

• Partial fraction decomposition: $\frac{1}{x^2 - 9} = \frac{1}{2\cdot3}\left(\frac{1}{x-3} - \frac{1}{x+3}\right)$

• Integral: $\int \frac{1}{x-a}\,dx = \ln|x-a| + C$

Step-by-Step Guidance

1. Decompose $\frac{1}{x^2 - 9}$ into partial fractions.

2. Integrate each term separately using the natural logarithm formula.

3. Set up the definite integral by evaluating the antiderivative at the upper and lower bounds, but do not compute the final value.

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Q11. Evaluate $\int_0^{3\Delta} \frac{-\sin x}{1 + \cos^2 x}\,dx$

Background

Topic: Substitution and Definite Integrals

This question tests your ability to use substitution to integrate a trigonometric function and evaluate a definite integral.

Key Terms and Formulas

• Let $u = \cos x$, then $du = -\sin x\,dx$

• Integral: $\int \frac{du}{1 + u^2} = \arctan(u) + C$

Step-by-Step Guidance

1. Let $u = \cos x$, so $du = -\sin x\,dx$.

2. Rewrite the integral in terms of $u$ and adjust the bounds accordingly.

3. Integrate using the formula for $\int \frac{du}{1 + u^2}$.

4. Set up the definite integral with the new bounds, but do not compute the final value.

Try solving on your own before revealing the answer!