Match an inner derivative
Evaluate $\int 3x^2(1+x^3)^5\,dx$.
Set $u=1+x^3$, so $du=3x^2\,dx$.
$$\int u^5\,du=\frac{u^6}{6}+C.$$
$$\therefore\ \int 3x^2(1+x^3)^5\,dx=\frac{(1+x^3)^6}{6}+C.$$
Differentiation checks the result.
Integration techniques
Recognize when substitution or integration by parts simplifies an integral, then carry the method through accurately.
By the end of this lesson
- choose between substitution and integration by parts
- complete a substitution including bounds
- apply integration by parts in a useful direction
Recognize the structure
Before calculating, decide what would make the integrand simpler. Is there an inside function together with its derivative? Try substitution. Is there a product with a factor that becomes simpler when differentiated? Try integration by parts.
Compare $2x\cos(x^2)$, where $2x$ is the derivative of the inside $x^2$, with $x e^x$, where differentiating $x$ removes that factor. These clues suggest a first move; they are not rules that make other valid methods impossible.
Substitution
Look for a composite function and the derivative of its inner expression.
$$\int f(g(x))g'(x)\,dx$$
Integration by parts
Look for a product where differentiating one factor simplifies it.
$$\int u\,dv=uv-\int v\,du$$
Checkpoint
Correct. $du=2x\,dx$ appears exactly, leaving $\int\cos u\,du$.
Not quite. Choose the method that removes the composite structure immediately.
Show explanation
$du=2x\,dx$ appears exactly, leaving $\int\cos u\,du$.
Example 1
Use substitution
Transform definite bounds
A substitution changes the coordinate used to describe the same endpoints. For $u=x^2$, the endpoints $x=1,3$ become $u=1,9$: the interval labels must follow the new variable.
With a definite integral, either change the bounds into $u$-values or return completely to $x$ before evaluating. Do not mix the two.
Checkpoint
Correct. Substitute each original $x$-bound into $u=x^2+1$.
Not quite. Bounds must be transformed by the same substitution as the integrand.
Show explanation
Substitute each original $x$-bound into $u=x^2+1$.
Example 2
Integrate by parts
Differentiate the algebraic factor
Evaluate $\int xe^x\,dx$.
Choose $u=x$ and $dv=e^x\,dx$. Then $du=dx$ and $v=e^x$.
$$\int xe^x\,dx=xe^x-\int e^x\,dx$$
$$=e^x(x-1)+C.$$
Differentiating $e^x(x-1)$ returns $xe^x$.
Order the method
Order a clean integration-by-parts method.
- Choose $u$ and $dv$
- Find $du$ and $v$
- Apply $uv-\int v\,du$
- Integrate the simpler remaining integral and check
Show a valid order
- Choose $u$ and $dv$
- Find $du$ and $v$
- Apply $uv-\int v\,du$
- Integrate the simpler remaining integral and check
The method now preserves the roles of $u$, $du$, $v$ and $dv$.
Combine methods only when needed
A substitution may reveal a product that then needs integration by parts. Make one structural simplification at a time.
Example 3
Use two stages
A logarithmic substitution
Evaluate $\int x\ln(x^2)\,dx$ for $x>0$.
Let $u=x^2$, so $du=2x\,dx$:
$$\int x\ln(x^2)\,dx=\frac12\int\ln u\,du.$$
Now integrate by parts: $\int\ln u\,du=u\ln u-u$.
$$\frac12\left[x^2\ln(x^2)-x^2\right]+C.$$
The condition $x>0$ keeps the logarithm defined.
Checkpoint
Correct. Differentiating $\ln x$ simplifies it to $1/x$, while $x$ integrates easily.
Not quite. There is no inner derivative pattern that removes the product by a simple substitution.
Show explanation
Differentiating $\ln x$ simplifies it to $1/x$, while $x$ integrates easily.
In the exam
Quick check
Your score is not saved.
Let $u=3x$, or account directly for the inner derivative.
Show explanation
Let $u=3x$, or account directly for the inner derivative.
The remaining integral should be simpler than the original.
Show explanation
The remaining integral should be simpler than the original.
A complete substitution uses one variable throughout.
Show explanation
A complete substitution uses one variable throughout.
All antiderivatives differ by a constant.
Show explanation
All antiderivatives differ by a constant.
The numerator is a constant multiple of the denominator’s derivative.
Show explanation
The numerator is a constant multiple of the denominator’s derivative.
Lesson Rewards
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