8.4. Integration by substitution [P3]

Math Response

Expand the power.

$$\int_{ }^{ }\left(x+1\right)^2dx=\int_{ }^{ }???\ dx$$

Type answer here

Correct Answer: x^2+2x+1

Deg^°

Rad

Math Response

Evaluate the integral.

$$\int_{ }^{ }\left(x+1\right)^2dx=\int_{ }^{ }x^2+2x+1\ dx=???$$

Type answer here

Correct Answer: \frac{1}{3}x^3+x^2+x+c

Deg^°

Rad

Consider!

8.4. Integration by substitution

Learning Objectives

At the end of the lesson, learners will be able to:

• Use a given substitution to simplify and evaluate integral.

Substitution method and Chain rule

Integration by substitution can be considered as the reverse process of differentiation by chain rule.

This method is used when a simple substitution can be applied that will transform a complicated integral into a simpler integral.

The integral must be completely rewritten in terms of the new variable.





In Mathematics Pure 3, the substitution will be given.

Substitution method

In the following example, we will see how we can use substitution method to simplify integration.

Substitution method

Math Response

Substitute $$t=x+1$$   to integrate $$\int_{ }^{ }\left(x+1\right)^7dx$$   .
Since $$t=x+1$$   , then $$\left(x+1\right)^7=???$$   .

Type answer here

Correct Answer: t^7

Deg^°

Rad

Math Response

Substitute $$t=x+1$$   to integrate $$\int_{ }^{ }\left(x+1\right)^7dx$$   .
Since $$t=x+1$$   , then $$\left(x+1\right)^7=t^7$$   .

So, $$\int_{ }^{ }\left(x+1\right)^7dx=\int_{ }^{ }???dx$$

Type answer here

Correct Answer: t^7

Deg^°

Rad

Substitution method

Math Response

Since $$t=x+1$$   , when we differentiate we have

$$\frac{dt}{dx}=???$$

Type answer here

Correct Answer: 1

Deg^°

Rad

Substitution method

Substitution method

Math Response

Since $$t=x+1$$   , then $$\left(x+1\right)^7=t^7$$   .
and also $$\frac{dt}{dx}=1$$   , so $$dx=dt$$   .

So, $$\int_{ }^{ }\left(x+1\right)^7dx=\int_{ }^{ }t^7dt=???$$ .

Type answer here

Correct Answer: \frac{1}{8}t^8+c

Deg^°

Rad

Math Response

Since $$t=x+1$$   , then $$\left(x+1\right)^7=t^7$$   .
and also $$\frac{dt}{dx}=1$$   , so $$dx=dt$$   .

So, $$\int_{ }^{ }\left(x+1\right)^7dx=\int_{ }^{ }t^7dt=\frac{1}{8}t^8+c=???$$ .

(the original question is in variable "x", so the answer should also be in "x")

Type answer here

Correct Answer: \frac{1}{8}\left(x+1\right)^8+c

Deg^°

Rad

Substitution method

Substitution method

Math Response

Since $$t=x^2-3$$ , then:

$$\frac{dt}{dx}=???$$

Type answer here

Correct Answer: 2x

Deg^°

Rad

Math Response

Since $$t=x^2-3$$ , then $$\frac{dt}{dx}=2x$$ .

Thus,

$$dx=???\ dt$$

Type answer here

Correct Answer: \frac{1}{2x}

Deg^°

Rad

Math Response

Since $$t=x^2-3$$ , then $$\frac{dt}{dx}=2x$$ .

Thus, $$dx=\frac{1}{2x}\ dt$$ .

So, the integral become:

$$\int_{ }^{ }\frac{x\ }{\sqrt[]{x^2-3}}dx=\int_{ }^{ }\frac{x\ }{\sqrt[]{???}}dx$$

Type answer here

Correct Answer: t

Deg^°

Rad

Math Response

Since $$t=x^2-3$$ , then $$\frac{dt}{dx}=2x$$ .

Thus, $$dx=\frac{1}{2x}\ dt$$ .

So, the integral become:

$$\int_{ }^{ }\frac{x\ }{\sqrt[]{x^2-3}}dx=\int_{ }^{ }\frac{x\ }{\sqrt[]{t}}\frac{1\ }{2x}dt=\int_{ }^{ }???dt$$

Type answer here

Correct Answer: \frac{1}{2\sqrt{t}}

Deg^°

Rad

Math Response

Since $$t=x^2-3$$ , then $$\frac{dt}{dx}=2x$$ .

Thus, $$dx=\frac{1}{2x}\ dt$$ .

So, the integral become:

$$\int_{ }^{ }\frac{x\ }{\sqrt[]{x^2-3}}dx=\int_{ }^{ }\frac{x\ }{\sqrt[]{t}}\frac{1\ }{2x}dt=\int_{ }^{ }\frac{1\ }{2\sqrt[]{t}}dt=\int_{ }^{ }\frac{1\ }{2}t^{-\frac{1}{2}}dt=???$$

Type answer here

Correct Answer: t^{\frac{1}{2}}+c

Deg^°

Rad

Math Response

Since $$t=x^2-3$$ , then $$\frac{dt}{dx}=2x$$ .

Thus, $$dx=\frac{1}{2x}\ dt$$ .

So, the integral become:

$$\int_{ }^{ }\frac{x\ }{\sqrt[]{x^2-3}}dx=\int_{ }^{ }\frac{x\ }{\sqrt[]{t}}\frac{1\ }{2x}dt=$$ $$\int_{ }^{ }\frac{1\ }{2\sqrt[]{t}}dt=\int_{ }^{ }\frac{1\ }{2}t^{-\frac{1}{2}}dt=t^{\frac{1}{2}}+c=???$$

Type answer here

Correct Answer: \left(x^2-3\right)^{\frac{1}{2}}+c

Deg^°

Rad