Pearl Carney

2021-11-19

Evaluate the integrals.

$\int x{(x+3)}^{10}dx$

Parminquale

Beginner2021-11-20Added 17 answers

Step 1

According to the question, we have to integrate the given integral$\int x{(x+3)}^{10}dx$ .

The given integral is indefinite integral and this type of integration have no fixed value that is why we add a constant value after the integration.

To solve the above integral, we have to use the power formula, which is given as follows,

$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$

Step 2

Rewrite the given integral,

$\int x{(x+3)}^{10}dx$

Now,

Let,$I=\int x{(x+3)}^{10}dx$ ...(1)

To solve further, we have to use substitution method, so substitutins x+3=t and proceeding as follows,

x+3=t...(1)

Differentiating both sides with respect to x, we get,

dx=dt

Now, substituting as x=(t−3) and dx=dt, in the equation (1), we get,

Step 3

$I=\int (t-3)\cdot {t}^{10}dt$

$=\int {t}^{11}dt-\int 3{t}^{10}dt$

$=\frac{{t}^{12}}{12}-\frac{3{t}^{11}}{11}+C$

Now, substitute back in the above answer as x+3=t,we get the final answer as,

$I=\frac{{(x+3)}^{12}}{12}-\frac{3{(x+3)}^{11}}{11}+C$

According to the question, we have to integrate the given integral

The given integral is indefinite integral and this type of integration have no fixed value that is why we add a constant value after the integration.

To solve the above integral, we have to use the power formula, which is given as follows,

Step 2

Rewrite the given integral,

Now,

Let,

To solve further, we have to use substitution method, so substitutins x+3=t and proceeding as follows,

x+3=t...(1)

Differentiating both sides with respect to x, we get,

dx=dt

Now, substituting as x=(t−3) and dx=dt, in the equation (1), we get,

Step 3

Now, substitute back in the above answer as x+3=t,we get the final answer as,

Sevensis1977

Beginner2021-11-21Added 15 answers

Step 1: Use Integration by Substitution.

Let u=x+3, du=dx then x dx=u-3du

Step 2: Using u and dudu above, rewrite$\int x{(x+3)}^{10}dx$ .

$\int (u-3){u}^{10}du$

Step 3: Expand.

$\int {u}^{11}-3{u}^{10}du$

Step 4: Use Power Rule:$\int {x}^{n}dx=\frac{{x}^{n+1}}{n+1}+C$ .

$\frac{{u}^{12}}{12}-\frac{3{u}^{11}}{11}$

Step 5: Substitute u=x+3 back into the original integral.

$\frac{{(x+3)}^{12}}{12}-\frac{3{(x+3)}^{11}}{11}$

Step 6: Add constant.

$\frac{{(x+3)}^{12}}{12}-\frac{3{(x+3)}^{11}}{11}+C$

Let u=x+3, du=dx then x dx=u-3du

Step 2: Using u and dudu above, rewrite

Step 3: Expand.

Step 4: Use Power Rule:

Step 5: Substitute u=x+3 back into the original integral.

Step 6: Add constant.