Math / Calculus

Questionfind $\int \frac{\ln (x+1)^{2}}{2 x+2} d x$

Studdy Solution

STEP 1

1. We are given the integral $\int \frac{\ln (x+1)^{2}}{2x+2} \, dx$ .
2. We will use substitution to simplify the integral.

STEP 2

1. Simplify the integrand.
2. Use substitution to simplify the integral.
3. Integrate the simplified expression.
4. Back-substitute to find the solution in terms of $x$ .

STEP 3

Simplify the integrand by recognizing that $\ln (x+1)^{2} = 2 \ln (x+1)$ :
$\int \frac{\ln (x+1)^{2}}{2x+2} \, dx = \int \frac{2 \ln (x+1)}{2(x+1)} \, dx$
Simplify further:
$= \int \frac{\ln (x+1)}{x+1} \, dx$

STEP 4

Use substitution. Let $u = \ln(x+1)$ , then $du = \frac{1}{x+1} \, dx$ .
Thus, the integral becomes:
$\int u \, du$

STEP 5

Integrate the simplified expression:
$\int u \, du = \frac{u^2}{2} + C$

STEP 6

Back-substitute $u = \ln(x+1)$ into the integrated expression:
$\frac{(\ln(x+1))^2}{2} + C$
The solution to the integral is:
$\boxed{\frac{(\ln(x+1))^2}{2} + C}$

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Questionfind $\int \frac{\ln (x+1)^{2}}{2 x+2} d x$

Studdy Solution

STEP 1

1. We are given the integral $\int \frac{\ln (x+1)^{2}}{2x+2} \, dx$ .
2. We will use substitution to simplify the integral.

STEP 2

1. Simplify the integrand.
2. Use substitution to simplify the integral.
3. Integrate the simplified expression.
4. Back-substitute to find the solution in terms of $x$ .

STEP 3

Simplify the integrand by recognizing that $\ln (x+1)^{2} = 2 \ln (x+1)$ :
$\int \frac{\ln (x+1)^{2}}{2x+2} \, dx = \int \frac{2 \ln (x+1)}{2(x+1)} \, dx$
Simplify further:
$= \int \frac{\ln (x+1)}{x+1} \, dx$

STEP 4

Use substitution. Let $u = \ln(x+1)$ , then $du = \frac{1}{x+1} \, dx$ .
Thus, the integral becomes:
$\int u \, du$

STEP 5

Integrate the simplified expression:
$\int u \, du = \frac{u^2}{2} + C$

STEP 6

Back-substitute $u = \ln(x+1)$ into the integrated expression:
$\frac{(\ln(x+1))^2}{2} + C$
The solution to the integral is:
$\boxed{\frac{(\ln(x+1))^2}{2} + C}$

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Questionfind ∫ln⁡(x+1)22x+2dx\int \frac{\ln (x+1)^{2}}{2 x+2} d x∫2x+2ln(x+1)2​dx

Studdy Solution

STEP 1

1. We are given the integral ∫ln⁡(x+1)22x+2 dx\int \frac{\ln (x+1)^{2}}{2x+2} \, dx∫2x+2ln(x+1)2​dx.2. We will use substitution to simplify the integral.

STEP 2

1. Simplify the integrand.2. Use substitution to simplify the integral.3. Integrate the simplified expression.4. Back-substitute to find the solution in terms of xxx.

STEP 3

STEP 4

Use substitution. Let u=ln⁡(x+1) u = \ln(x+1) u=ln(x+1), then du=1x+1 dx du = \frac{1}{x+1} \, dx du=x+11​dx.Thus, the integral becomes:∫u du\int u \, du∫udu

STEP 5

Integrate the simplified expression:∫u du=u22+C\int u \, du = \frac{u^2}{2} + C∫udu=2u2​+C

STEP 6

Back-substitute u=ln⁡(x+1) u = \ln(x+1) u=ln(x+1) into the integrated expression:(ln⁡(x+1))22+C\frac{(\ln(x+1))^2}{2} + C2(ln(x+1))2​+C The solution to the integral is:(ln⁡(x+1))22+C\boxed{\frac{(\ln(x+1))^2}{2} + C}2(ln(x+1))2​+C​

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Questionfind $\int \frac{\ln (x+1)^{2}}{2 x+2} d x$

1. We are given the integral $\int \frac{\ln (x+1)^{2}}{2x+2} \, dx$ .
2. We will use substitution to simplify the integral.

1. Simplify the integrand.
2. Use substitution to simplify the integral.
3. Integrate the simplified expression.
4. Back-substitute to find the solution in terms of $x$ .

Use substitution. Let $u = \ln(x+1)$ , then $du = \frac{1}{x+1} \, dx$ .
Thus, the integral becomes:
$\int u \, du$

Integrate the simplified expression:
$\int u \, du = \frac{u^2}{2} + C$

Back-substitute $u = \ln(x+1)$ into the integrated expression:
$\frac{(\ln(x+1))^2}{2} + C$
The solution to the integral is:
$\boxed{\frac{(\ln(x+1))^2}{2} + C}$