Find out the answer to "Evaluate the integral and check answer by differentiating...."

parthe0o

Answered question

2021-11-10

Evaluate the integral and check answer by differentiating.

\int x (1 + x^{3}) d x

Answer & Explanation

mylouscrapza

Beginner2021-11-11Added 22 answers

Step 1
We have the integral :

\int x (1 + x^{3}) d x

We have to integrate this integral and then we have to check our answer by differentiating.
Step 2
Let

g (x) = x (1 + x^{3})

We have the integral :

\int x (1 + x^{3}) d x

= \int (x + x^{4}) d x

= \int x d x + \int x^{4} d x

= \frac{x^{2}}{2} + \frac{x^{5}}{5} + C

where C is an arbitrary constant.
This gives the required value of the integral.
Step 3
Now we have to check by differentiating whether this integral is correct or not .
Let

f (x) = \frac{x^{2}}{2} + \frac{x^{5}}{5} + C

Now differentiating both sides with respect to x we get :

f^{'} (x) = \frac{1}{2} \times 2 x + \frac{1}{5} \times 5 x^{4} + 0

= x + x^{4}

= x (1 + x^{3}) = g (x)

Thus we ensure that our evaluation of the integral is correct.

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