"Determine the following indefinite integral. ∫10t5−3tdt" was answered by our community

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hadejada7x

Answered question

2021-12-10

Determine the following indefinite integral.
$\int \frac{10 t^{5} - 3}{t} d t$

Answer & Explanation

Juan Spiller

Beginner2021-12-11Added 38 answers

Step 1
Given: $\int \frac{10 t^{5} - 3}{t} d t$
for evaluating this integral we separate terms, then integrate it
Step 2
so,
$\int \frac{10 t^{5} - 3}{t} d t = \int (\frac{10 t^{5}}{t} - \frac{3}{t}) d t$
$= 10 \int t^{4} d t - 3 \int \frac{d t}{t}$
$(∵ \int x^{m} d x = \frac{x^{m + 1}}{m + 1} + c, \int \frac{d x}{x} = \ln + c)$
$= 10 (\frac{t^{5}}{5}) - 3 (\ln t) + c$
$= 2 t^{5} - 3 \ln t + c$
so, given integral is equal to $(2 t^{5} - 3 \ln t + c)$
now differentiating integral value for checking
$y = 2 t^{5} - 3 \ln t + c$
so,
$\frac{d y}{d t} = 2 (5) t^{4} - \frac{3}{t} + 0$
$(∵ \frac{d}{d x} (k x^{n}) = k n x^{n - 1}, \frac{d}{d x} (\ln x) = \frac{1}{x})$
$= 10 t^{4} - \frac{3}{t}$
$= \frac{10 t^{5} - 3}{t}$
hence, given integral is equal to $2 t^{5} - 3 \ln t + c$ .

Philip Williams

Beginner2021-12-12Added 39 answers

Given:
$\int \frac{10 t^{5} - 3}{t} d t$
Separate the fraction
$\int \frac{10 t^{5}}{t} - \frac{3}{t} d t$
Simplify the expression
$\int 10 t^{4} - \frac{3}{t} d t$
$\int 10 t^{4} - \int \frac{3}{t} d t$
Evaluate
$2 t^{5} - 3 \ln (| t |)$
Add C
Answer:
$2 t^{5} - 3 \ln (| t |) + C$

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