Question Statement

Evaluate the definite integral:

${\int }_1^4\frac{\cos \sqrt{x}}{2\sqrt{x}}dx$

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Background and Explanation

This problem requires the method of substitution (u-substitution) for definite integrals. The key insight is recognizing that the integrand contains a composite function $\cos (\sqrt{x})$ multiplied by the derivative of the inner function $\frac{1}{2\sqrt{x}}$, which allows for a natural change of variables to simplify the integral.

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Solution

Let $I$ denote the given integral:

$I={\int }_1^4\frac{\cos \sqrt{x}}{2\sqrt{x}}dx$

Step 1: Choose the substitution

Observe that the derivative of $\sqrt{x}$ is $\frac{1}{2\sqrt{x}}$, which appears as a factor in the integrand. This suggests the substitution:

$t=\sqrt{x}$

Step 2: Transform the limits of integration

When changing variables, we must convert the $x$-limits to $t$-limits:

• When $x=1$: $t=\sqrt{1}=1$
• When $x=4$: $t=\sqrt{4}=2$

Step 3: Compute the differential

Differentiating $t=\sqrt{x}$ with respect to $x$:

$\frac{dt}{dx}=\frac{1}{2\sqrt{x}}$

Rearranging gives:

$\frac{1}{2\sqrt{x}}dx=dt$

Step 4: Rewrite and evaluate the integral

Substituting $t=\sqrt{x}$, $dt=\frac{1}{2\sqrt{x}}dx$, and the new limits $t=1$ to $t=2$:

$\begin{array}{rl}I& ={\int }_1^2\cos tdt\\ & ={\left[\sin t\right]}_1^2\\ & =\sin 2-\sin 1\end{array}$

Therefore, the value of the integral is $\sin 2-\sin 1$.

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Key Formulas or Methods Used

• Substitution Rule for Definite Integrals: ${\int }_a^{b}f(g(x))g^{\prime }(x)dx={\int }_{g(a)}^{g(b)}f(u)du$
• Power Rule for Differentiation: $\frac{d}{dx}(\sqrt{x})=\frac{d}{dx}(x^{1/2})=\frac{1}{2\sqrt{x}}$
• Basic Trigonometric Integral: $\int \cos (t)dt=\sin (t)+C$
• Fundamental Theorem of Calculus: ${\int }_a^{b}f(x)dx=F(b)-F(a)$, where $F$ is an antiderivative of $f$

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Summary of Steps

1. Identify substitution: Let $t=\sqrt{x}$ since $\frac{1}{2\sqrt{x}}$ (the derivative of $\sqrt{x}$) appears in the integrand.
2. Change limits: Convert $x=1$ to $t=1$ and $x=4$ to $t=2$.
3. Find differential: Establish that $dt=\frac{1}{2\sqrt{x}}dx$.
4. Transform integral: Rewrite as ${\int }_1^2\cos (t)dt$.
5. Integrate: Find antiderivative $\sin (t)$ and evaluate at bounds.
6. Final evaluation: Compute $\sin (2)-\sin (1)$.