Math / Calculus

Question4. $\int_{0}^{11-x} d x=$ (a) $\frac{1}{2} \ln 2$ (b) $2 \ln 2+1$ (c) $2 \ln 2-1$ (d) $\frac{1}{2} \ln 2-1$

Studdy Solution

STEP 1

1. The integral to solve is $\int_{0}^{1} \frac{1}{1+x} \, dx$ .
2. This is a definite integral involving a natural logarithm function.

STEP 2

1. Identify the integral form.
2. Evaluate the integral.
3. Apply the limits of integration.
4. Simplify the result.

STEP 3

Recognize that the integral $\int \frac{1}{1+x} \, dx$ is the natural logarithm function:
$\int \frac{1}{1+x} \, dx = \ln|1+x| + C$

STEP 4

Evaluate the integral from 0 to 1:
$\int_{0}^{1} \frac{1}{1+x} \, dx = \left[ \ln|1+x| \right]_{0}^{1}$

STEP 5

Apply the limits of integration:
$\left[ \ln|1+x| \right]_{0}^{1} = \ln|1+1| - \ln|1+0|$
$= \ln 2 - \ln 1$

STEP 6

Simplify the result:
Since $\ln 1 = 0$ , we have:
$\ln 2 - 0 = \ln 2$
The value of the integral is:
$\boxed{\ln 2}$
However, none of the provided options match $\ln 2$ directly. Please verify the problem statement or options.

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Question4. $\int_{0}^{11-x} d x=$ (a) $\frac{1}{2} \ln 2$ (b) $2 \ln 2+1$ (c) $2 \ln 2-1$ (d) $\frac{1}{2} \ln 2-1$

Studdy Solution

STEP 1

1. The integral to solve is $\int_{0}^{1} \frac{1}{1+x} \, dx$ .
2. This is a definite integral involving a natural logarithm function.

STEP 2

1. Identify the integral form.
2. Evaluate the integral.
3. Apply the limits of integration.
4. Simplify the result.

STEP 3

Recognize that the integral $\int \frac{1}{1+x} \, dx$ is the natural logarithm function:
$\int \frac{1}{1+x} \, dx = \ln|1+x| + C$

STEP 4

Evaluate the integral from 0 to 1:
$\int_{0}^{1} \frac{1}{1+x} \, dx = \left[ \ln|1+x| \right]_{0}^{1}$

STEP 5

Apply the limits of integration:
$\left[ \ln|1+x| \right]_{0}^{1} = \ln|1+1| - \ln|1+0|$
$= \ln 2 - \ln 1$

STEP 6

Simplify the result:
Since $\ln 1 = 0$ , we have:
$\ln 2 - 0 = \ln 2$
The value of the integral is:
$\boxed{\ln 2}$
However, none of the provided options match $\ln 2$ directly. Please verify the problem statement or options.

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Question4. ∫011−xdx=\int_{0}^{11-x} d x=∫011−x​dx= (a) 12ln⁡2\frac{1}{2} \ln 221​ln2 (b) 2ln⁡2+12 \ln 2+12ln2+1 (c) 2ln⁡2−12 \ln 2-12ln2−1 (d) 12ln⁡2−1\frac{1}{2} \ln 2-121​ln2−1

Studdy Solution

STEP 1

1. The integral to solve is ∫0111+x dx\int_{0}^{1} \frac{1}{1+x} \, dx∫01​1+x1​dx.2. This is a definite integral involving a natural logarithm function.

STEP 2

1. Identify the integral form.2. Evaluate the integral.3. Apply the limits of integration.4. Simplify the result.

STEP 3

Recognize that the integral ∫11+x dx\int \frac{1}{1+x} \, dx∫1+x1​dx is the natural logarithm function:∫11+x dx=ln⁡∣1+x∣+C \int \frac{1}{1+x} \, dx = \ln|1+x| + C ∫1+x1​dx=ln∣1+x∣+C

STEP 4

Evaluate the integral from 0 to 1:∫0111+x dx=[ln⁡∣1+x∣]01 \int_{0}^{1} \frac{1}{1+x} \, dx = \left[ \ln|1+x| \right]_{0}^{1} ∫01​1+x1​dx=[ln∣1+x∣]01​

STEP 5

Apply the limits of integration:[ln⁡∣1+x∣]01=ln⁡∣1+1∣−ln⁡∣1+0∣ \left[ \ln|1+x| \right]_{0}^{1} = \ln|1+1| - \ln|1+0| [ln∣1+x∣]01​=ln∣1+1∣−ln∣1+0∣=ln⁡2−ln⁡1 = \ln 2 - \ln 1 =ln2−ln1

STEP 6

Simplify the result:Since ln⁡1=0\ln 1 = 0ln1=0, we have:ln⁡2−0=ln⁡2 \ln 2 - 0 = \ln 2 ln2−0=ln2The value of the integral is:ln⁡2 \boxed{\ln 2} ln2​However, none of the provided options match ln⁡2\ln 2ln2 directly. Please verify the problem statement or options.

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Question4. $\int_{0}^{11-x} d x=$ (a) $\frac{1}{2} \ln 2$ (b) $2 \ln 2+1$ (c) $2 \ln 2-1$ (d) $\frac{1}{2} \ln 2-1$

1. The integral to solve is $\int_{0}^{1} \frac{1}{1+x} \, dx$ .
2. This is a definite integral involving a natural logarithm function.

1. Identify the integral form.
2. Evaluate the integral.
3. Apply the limits of integration.
4. Simplify the result.

Recognize that the integral $\int \frac{1}{1+x} \, dx$ is the natural logarithm function:
$\int \frac{1}{1+x} \, dx = \ln|1+x| + C$

Evaluate the integral from 0 to 1:
$\int_{0}^{1} \frac{1}{1+x} \, dx = \left[ \ln|1+x| \right]_{0}^{1}$

Apply the limits of integration:
$\left[ \ln|1+x| \right]_{0}^{1} = \ln|1+1| - \ln|1+0|$
$= \ln 2 - \ln 1$

Simplify the result:
Since $\ln 1 = 0$ , we have:
$\ln 2 - 0 = \ln 2$
The value of the integral is:
$\boxed{\ln 2}$
However, none of the provided options match $\ln 2$ directly. Please verify the problem statement or options.