Answer:
1. The integral of
is
, where
is the constant of integration.
2. The integral of
can be rewritten using the double-angle formula as

3. The integral of
is
.
4. The integral of
is
.
5. The integral of
is
.
Explanation:
Let's break down each integral:
1. Integral of

This is a case of a simple substitution. We can let
, then
. Substituting these into the integral, we get
, which is simply
. Substituting back for
, we get
.
2. Integral of

The double-angle formula is used here. We know that
. Substituting this into the integral, we get
, which can be separated into two simpler integrals:
. The integral of
, and the integral of
is
. So, the result is
.
3. Integral of \(\frac{\cos(\sqrt{x})}{\sqrt{x}} dx\)
This is another case of simple substitution. We can let
, then
. Substituting these into the integral, we get
. Substituting back for
, we get
.
4. Integral of
![\(\frac{1}{\sqrt[3]{1-7x}} dx\)](https://img.qammunity.org/2024/formulas/mathematics/college/aanqax5n69selva9pipbf4pim6niqxe08w.png)
Here, we can let
, then
, or
. Substituting these into the integral, we get
, which is
. Substituting back for
, we get
.
5. Integral of

This is a standard form of integral that results in a natural logarithm. The integral of
is
. So, the result is
.
Hope This Helps!