e - 1
Evaluating Definite Integrals using Substitution
The question asks us to evaluate the definite integral \( \mathop \smallint \limits_0^{\frac{\pi }{2}} {e^{\sin x}}\cos x\;dx \). This integral involves the product of a composite function \( e^{\sin x} \) and another function \( \cos x \). This structure often suggests using a substitution method for integration.
Let's analyze the integrand: \( e^{\sin x}\cos x \). We observe that the derivative of the inner function \( \sin x \) is \( \cos x \), which is present in the integrand. This makes the substitution \( u = \sin x \) a suitable approach.
Step-by-Step Definite Integral Evaluation
We will use the substitution method to solve this definite integral problem.
- Identify the Substitution: Let \( u = \sin x \).
- Find the Differential \( du \): Differentiating both sides with respect to \( x \), we get \( \frac{du}{dx} = \cos x \). This implies \( du = \cos x\;dx \).
-
Change the Limits of Integration:
Since it is a definite integral, we need to change the limits from \( x \) values to \( u \) values.
- Lower limit: When \( x = 0 \), \( u = \sin(0) = 0 \).
- Upper limit: When \( x = \frac{\pi}{2} \), \( u = \sin(\frac{\pi}{2}) = 1 \).
- Rewrite the Integral in terms of \( u \): The original integral is \( \mathop \smallint \limits_0^{\frac{\pi }{2}} {e^{\sin x}}\cos x\;dx \). Substituting \( u = \sin x \) and \( du = \cos x\;dx \), and using the new limits, the integral becomes: \( \mathop \smallint \limits_{u=0}^{u=1} {e^{u}}\;du \)
- Evaluate the New Integral: The integral of \( e^u \) with respect to \( u \) is \( e^u \). Now we evaluate this from the lower limit \( u=0 \) to the upper limit \( u=1 \). \( \mathop \smallint \limits_{0}^{1} {e^{u}}\;du = \left[ e^u \right]_0^1 \)
- Apply the Limits: \( \left[ e^u \right]_0^1 = e^1 - e^0 \) We know that \( e^1 = e \) and \( e^0 = 1 \) (any non-zero number raised to the power of 0 is 1). So, \( e^1 - e^0 = e - 1 \).
Thus, the value of the definite integral \( \mathop \smallint \limits_0^{\frac{\pi }{2}} {e^{\sin x}}\cos x\;dx \) is \( e - 1 \).
Understanding the Options
Let's compare our calculated value with the given options:
- Option 1: \( e + 1 \)
- Option 2: \( e - 1 \)
- Option 3: \( e + 2 \)
- Option 4: \( e \)
Our result, \( e - 1 \), matches Option 2.
| Original Integral | Substitution | New Integral (with limits) | Result |
|---|---|---|---|
| \( \mathop \smallint \limits_0^{\frac{\pi }{2}} {e^{\sin x}}\cos x\;dx \) | \( u = \sin x \) \( du = \cos x\;dx \) Limits: \( x=0 \to u=0 \), \( x=\frac{\pi}{2} \to u=1 \) |
\( \mathop \smallint \limits_{0}^{1} {e^{u}}\;du \) | \( e - 1 \) |
Revision Table: Key Concepts
| Concept | Description |
|---|---|
| Definite Integral | An integral with upper and lower limits, representing the net signed area under a curve. |
| Substitution Method | A technique used to simplify integrals by replacing a variable with a function of a new variable. Essential when the integrand contains a function and its derivative. |
| Changing Limits | When using substitution in a definite integral, the original limits must be transformed according to the substitution formula for the new variable. |
| Fundamental Theorem of Calculus | States that if \( F(x) \) is an antiderivative of \( f(x) \), then \( \mathop \smallint \limits_a^b f(x)\;dx = F(b) - F(a) \). |
Additional Information: Integral of \( e^u \) and \( \sin x \)
The integral of \( e^u \) is one of the most fundamental integrals. \( \mathop \smallint e^u\;du = e^u + C \), where \( C \) is the constant of integration for indefinite integrals.
The trigonometric function \( \sin x \) has a derivative of \( \cos x \). This property was crucial in applying the substitution \( u = \sin x \) effectively in this definite integral problem. The substitution simplifies the integrand significantly, making the integral easy to evaluate.
Remember that when performing substitution in definite integrals, it's generally easier to change the limits of integration to the new variable rather than changing back to the original variable after integration.
Similar Questions
What is the value of \(8 I_1^2\)
What is the value of I2 ?
What is \(\mathop \smallint \nolimits_{{{\rm{e}}^{ - 1}}}^{{{\rm{e}}^2}} \left| {\frac{{\ln {\rm{x}}}}{{\rm{x}}}} \right|{\rm{dx}}\) equal to?
What is \(\mathop \smallint \limits_{ - 2}^2 {\rm{x\;dx}} - \mathop \smallint \limits_{ - 2}^2 \left[ {\rm{x}} \right]{\rm{dx}}\) equal to, where [⋅] is the greatest integer function?
What is \(\displaystyle\int_0^1 \ln \left(\frac{1}{x}−1\right)\) dx equal to ?
- If \(\displaystyle\int_0^{\pi/2}\) (sin 4 x + cos 4 x)dx = k, then what is the value of \(\displaystyle\int_0^{20 \pi}\) (sin 4 x + cos 4 x)dx ?
- What is \(\displaystyle\int_{−\pi/2}^{\pi/2}\) (e cos x sin x + e sin x cos x)dx equal to ?
What is \(\rm \int^\pi _0 ln\left(tan\frac{x}{2}\right) dx\) equal to?
What is the area bounded by y = [x], where [⋅] is the greatest integer function, the x-axis and the lines x = -1.5 and x = -1.8?
- \(\mathop \smallint \limits_0^{\frac{\pi }{2}} |\sin x - \cos x|dx\) is equal to
Important Questions from Definite Integrals
The Legendre polynomials P n(x), n = 0, 1, 2, ..., satisfying the orthogonailty condition \(\int_{{\rm{ - 1}}}^{\rm{1}} {{{\rm{P}}_{\rm{n}}}\left( {\rm{x}} \right){{\rm{P}}_{\rm{m}}}} \left( {\rm{x}} \right){\rm{dx}}\,{\rm{ = }}\,\frac{{\rm{2}}}{{{\rm{2n + 1}}}}{{\rm{\delta }}_{{\rm{nm}}}}\) on the interval [-1, +1], may be defined by the Rodrigues formula P n(x) = \(\frac{{\rm{1}}}{{{{\rm{2}}^{\rm{n}}}{\rm{n!}}}}\frac{{{{\rm{d}}^{\rm{n}}}}}{{{\rm{d}}{{\rm{x}}^{\rm{n}}}}}{\left( {{{\rm{x}}^{\rm{2}}}{\rm{ - 1}}} \right)^{\rm{n}}}\) . The value of the definite integral \(\int_{{\rm{ - 1}}}^{\rm{1}} {\left( {{\rm{4 + 2x - 3}}{{\rm{x}}^{\rm{2}}}{\rm{ + 4}}{{\rm{x}}^{\rm{3}}}} \right){{\rm{P}}_{\rm{3}}}\left( {\rm{x}} \right){\rm{dx}}} \) is
- \(\rm \displaystyle\int_1^3 (e^{\log x} + 1) dx\) is equal to
A parametric curve is defined \(x = cos\left(\frac{\Pi t}{2}\right) , Y= sin\left(\frac{\Pi t}{2}\right)\) in the range of \(0\leq t\leq 1\) . It is rotated about X-axis by 360°.
‘What is the area of the surface generated?
if \(\displaystyle\int\dfrac{\sin x}{\sin (x-a)}dx=Ax+B\log |sin(x-a)|+ C\) where A, B and c are real constants then:
Which of the following is NOT a property of definite integral?
