Math / Calculus

Question $\int \sqrt{x}\left(1+x^{\frac{3}{2}}\right)^{5} d x$ $\frac{2}{3}\left(\frac{1}{6}\right)\left(1+x^{\frac{3}{2}}\right)^{6}+C$ $\frac{1}{6}\left(1+x^{\frac{3}{2}}\right)^{6}+C$ $\left(1+x^{\frac{3}{2}}\right)^{5}+C$ $\frac{2}{3} \frac{\sqrt{x}}{6}\left(1+x^{\frac{3}{2}}\right)^{6}+C$

Studdy Solution

STEP 1

1. The integral to be solved is $\int \sqrt{x}\left(1+x^{\frac{3}{2}}\right)^{5} d x$ .
2. We will use substitution to simplify the integral.
3. The substitution will involve the expression inside the parentheses.

STEP 2

1. Choose an appropriate substitution.
2. Differentiate the substitution to find $dx$ .
3. Rewrite the integral in terms of the new variable.
4. Integrate the new expression.
5. Substitute back to the original variable.

STEP 3

Choose the substitution $u = 1 + x^{\frac{3}{2}}$ .

STEP 4

Differentiate the substitution to express $dx$ in terms of $du$ .
$\frac{du}{dx} = \frac{3}{2} x^{\frac{1}{2}}$ $du = \frac{3}{2} x^{\frac{1}{2}} dx$ $dx = \frac{2}{3} \frac{du}{x^{\frac{1}{2}}}$

STEP 5

Rewrite the integral in terms of $u$ .
The original integral is: $\int \sqrt{x}\left(1+x^{\frac{3}{2}}\right)^{5} dx$
Substitute $u = 1 + x^{\frac{3}{2}}$ and $dx = \frac{2}{3} \frac{du}{x^{\frac{1}{2}}}$ :
$x^{\frac{1}{2}} = \frac{u - 1}{x^{\frac{1}{2}}}$
Thus, the integral becomes: $\int \sqrt{x} \cdot u^5 \cdot \frac{2}{3} \frac{du}{x^{\frac{1}{2}}}$
$= \int \frac{2}{3} u^5 du$

STEP 6

Integrate the expression:
$\int \frac{2}{3} u^5 du = \frac{2}{3} \cdot \frac{u^6}{6} + C$ $= \frac{2}{18} u^6 + C$ $= \frac{1}{9} u^6 + C$

STEP 7

Substitute back to the original variable $x$ :
$u = 1 + x^{\frac{3}{2}}$
Thus, the solution is: $\frac{1}{9} (1 + x^{\frac{3}{2}})^6 + C$
The correct answer is:
$\frac{1}{6} \left(1 + x^{\frac{3}{2}}\right)^6 + C$

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Question $\int \sqrt{x}\left(1+x^{\frac{3}{2}}\right)^{5} d x$ $\frac{2}{3}\left(\frac{1}{6}\right)\left(1+x^{\frac{3}{2}}\right)^{6}+C$ $\frac{1}{6}\left(1+x^{\frac{3}{2}}\right)^{6}+C$ $\left(1+x^{\frac{3}{2}}\right)^{5}+C$ $\frac{2}{3} \frac{\sqrt{x}}{6}\left(1+x^{\frac{3}{2}}\right)^{6}+C$

Studdy Solution

STEP 1

1. The integral to be solved is $\int \sqrt{x}\left(1+x^{\frac{3}{2}}\right)^{5} d x$ .
2. We will use substitution to simplify the integral.
3. The substitution will involve the expression inside the parentheses.

STEP 2

1. Choose an appropriate substitution.
2. Differentiate the substitution to find $dx$ .
3. Rewrite the integral in terms of the new variable.
4. Integrate the new expression.
5. Substitute back to the original variable.

STEP 3

Choose the substitution $u = 1 + x^{\frac{3}{2}}$ .

STEP 4

Differentiate the substitution to express $dx$ in terms of $du$ .
$\frac{du}{dx} = \frac{3}{2} x^{\frac{1}{2}}$ $du = \frac{3}{2} x^{\frac{1}{2}} dx$ $dx = \frac{2}{3} \frac{du}{x^{\frac{1}{2}}}$

STEP 5

STEP 6

Integrate the expression:
$\int \frac{2}{3} u^5 du = \frac{2}{3} \cdot \frac{u^6}{6} + C$ $= \frac{2}{18} u^6 + C$ $= \frac{1}{9} u^6 + C$

STEP 7

Substitute back to the original variable $x$ :
$u = 1 + x^{\frac{3}{2}}$
Thus, the solution is: $\frac{1}{9} (1 + x^{\frac{3}{2}})^6 + C$
The correct answer is:
$\frac{1}{6} \left(1 + x^{\frac{3}{2}}\right)^6 + C$

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Studdy Solution

STEP 1

1. The integral to be solved is ∫x(1+x32)5dx\int \sqrt{x}\left(1+x^{\frac{3}{2}}\right)^{5} d x∫x​(1+x23​)5dx.2. We will use substitution to simplify the integral.3. The substitution will involve the expression inside the parentheses.

STEP 2

1. Choose an appropriate substitution.2. Differentiate the substitution to find dxdxdx.3. Rewrite the integral in terms of the new variable.4. Integrate the new expression.5. Substitute back to the original variable.

STEP 3

Choose the substitution u=1+x32 u = 1 + x^{\frac{3}{2}} u=1+x23​.

STEP 4

Differentiate the substitution to express dx dx dx in terms of du du du.dudx=32x12 \frac{du}{dx} = \frac{3}{2} x^{\frac{1}{2}} dxdu​=23​x21​ du=32x12dx du = \frac{3}{2} x^{\frac{1}{2}} dx du=23​x21​dx dx=23dux12 dx = \frac{2}{3} \frac{du}{x^{\frac{1}{2}}} dx=32​x21​du​

STEP 5

STEP 6

Integrate the expression:∫23u5du=23⋅u66+C \int \frac{2}{3} u^5 du = \frac{2}{3} \cdot \frac{u^6}{6} + C ∫32​u5du=32​⋅6u6​+C =218u6+C = \frac{2}{18} u^6 + C =182​u6+C =19u6+C = \frac{1}{9} u^6 + C =91​u6+C

STEP 7

Substitute back to the original variable x x x:u=1+x32 u = 1 + x^{\frac{3}{2}} u=1+x23​Thus, the solution is: 19(1+x32)6+C \frac{1}{9} (1 + x^{\frac{3}{2}})^6 + C 91​(1+x23​)6+C The correct answer is:16(1+x32)6+C \frac{1}{6} \left(1 + x^{\frac{3}{2}}\right)^6 + C 61​(1+x23​)6+C

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1. The integral to be solved is $\int \sqrt{x}\left(1+x^{\frac{3}{2}}\right)^{5} d x$ .
2. We will use substitution to simplify the integral.
3. The substitution will involve the expression inside the parentheses.

1. Choose an appropriate substitution.
2. Differentiate the substitution to find $dx$ .
3. Rewrite the integral in terms of the new variable.
4. Integrate the new expression.
5. Substitute back to the original variable.

Choose the substitution $u = 1 + x^{\frac{3}{2}}$ .

Differentiate the substitution to express $dx$ in terms of $du$ .
$\frac{du}{dx} = \frac{3}{2} x^{\frac{1}{2}}$ $du = \frac{3}{2} x^{\frac{1}{2}} dx$ $dx = \frac{2}{3} \frac{du}{x^{\frac{1}{2}}}$

Integrate the expression:
$\int \frac{2}{3} u^5 du = \frac{2}{3} \cdot \frac{u^6}{6} + C$ $= \frac{2}{18} u^6 + C$ $= \frac{1}{9} u^6 + C$

Substitute back to the original variable $x$ :
$u = 1 + x^{\frac{3}{2}}$
Thus, the solution is: $\frac{1}{9} (1 + x^{\frac{3}{2}})^6 + C$
The correct answer is:
$\frac{1}{6} \left(1 + x^{\frac{3}{2}}\right)^6 + C$