/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 17 \(\int \frac{t d t}{\sqrt{t+3}}\... [FREE SOLUTION] | 91影视

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Chapter 6: Problem 17

\(\int \frac{t d t}{\sqrt{t+3}}\)

Short Answer

Expert verified
The integral is \( \frac{2}{3} (t+3)^{3/2} - 6 (t+3)^{1/2} + C \).

Step by step solution

01

- Substitution

Let us use the substitution method. Set \( u = t + 3 \), then \( du = dt \).
02

- Substitute and Rewrite the Integral

Rewrite the integral in terms of \( u \).\( t = u - 3 \). Hence the integral becomes: \( \int \frac{(u-3) du}{\sqrt{u}} \).
03

- Simplify the Integral

Divide the terms inside the integral:\( \int (u^{1/2} - 3 u^{-1/2}) du \).
04

- Integrate Each Term

Integrate the terms separately:\( \int u^{1/2} du - 3 \int u^{-1/2} du \).The integrals are:\( \int u^{1/2} du = \frac{2}{3} u^{3/2} \)and\( \int u^{-1/2} du = 2 u^{1/2} \). Therefore, the result becomes:\( \frac{2}{3} u^{3/2} - 6 u^{1/2} \).
05

- Substitute Back in Terms of t

Replace \( u \) with \( t + 3 \):\( \frac{2}{3} (t+3)^{3/2} - 6 (t+3)^{1/2} + C \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

indefinite integration
Indefinite integration, often referred to simply as integration, is the process of finding an antiderivative of a function. This means we are looking for a function whose derivative will give us the original function. The general form for an indefinite integral is denoted by \(\textstyle \int f(x)dx\) and the result will include a constant of integration, usually denoted as \

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