# Integrate the function \int \frac{1}{(20+2t)} dt

## Question:

Integrate the function {eq}\int \frac{1}{(20+2t)} dt {/eq}

## Integration:

Integration is the opposite of differentiation. Here, we have to compute the integral of the function.

In mathematics, there are many techniques to solve the integral. One such technique is integration by substitution.

Some integrals can be easily solved by choosing a suitable substitution which in turn changes the function into something that can be easily integrated.

Given

{eq}\int \dfrac{1}{(20+2t)} dt {/eq}

We have to integrate the function.

{eq}\int \dfrac{1}{(20+2t)} dt=\int \dfrac{1}{2(10+t)}dt {/eq}

We use the substitution {eq}10+t=u {/eq} to solve the integral.

Differentiate both sides of the substitution with respect to {eq}t {/eq}

{eq}\begin{align} 10+t &=u\\ \dfrac{\mathrm{d} }{\mathrm{d} t}(10+t) &=\dfrac{\mathrm{d} u}{\mathrm{d} t}\\ 0+1 &= \dfrac{\mathrm{d} u}{\mathrm{d} t}\\ dt &=du \end{align} {/eq}

Applying this substitution, we have:

{eq}\begin{align} \int \dfrac{1}{(20+2t)} dt &=\int \dfrac{1}{2(10+t)}dt\\ &=\dfrac{1}{2}\int \dfrac{du}{u}\\ &=\dfrac{1}{2}\ln \left | u \right |+C \end{align} {/eq}

Reversing the substitution, we have:

{eq}\color{blue}{\int \dfrac{1}{(20+2t)} dt=\dfrac{1}{2}\ln \left | 10+t \right |+C} {/eq}

How to Solve Integrals Using Substitution

from Math 104: Calculus

Chapter 13 / Lesson 5
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